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Individually hand made excellence. More plate area than other discus. Meets all NCAA specifications. Five Star world class high spin high school teens professional level Recently thrown for over feet. There are 4 Xing-Yi forms: This is the 2nd tape of 2 parts. Smooth glass cooktop features 5 elements PreciseSelect offers 17 different direct cooking level settings The bridge elements accomodates oblong or oversized pans and griddles The two level heat indica Extra-wide, adjustable comfort cuffs give these 15 in. Primarily designed for lower body resistance training. Use for lateral shuffles, leg extensions and If its tan is positive and sin negative?
If its cot is negative and cos negative? If its csc is negative and cot positive? If its cos is positive and tan negative? If its sec is negative and tan negative? A railroad embankment is 9 ft. If a railroad embankment is 7 ft. Make up a similar example for yourself. Similar results are readily perceived for other quadrants by the use of a figure showing the sines as lines in the different quadrants. Similarly in the first quadrant Fig. Formulas for the Acute Angle extended to any Angle. The equations and formulas proved in Arts. Thus, on each of the Figs.
OMI, or tan x: For if angle XOP of Figs. P Also t ord. P cosx cosx dist. One function of an angle being given, the other functions may be found in a manner similar to that used in Art. By the table of signs Art. The positive tangent occurs see Art. Find the values of sin, cos, tan, and cot of the following angles: Verify geometrically the results obtained in Exs. The same results may readily be obtained for angles ending in the second, third, and fourth quadrants by use of the following diagrams.
Simplify the following expressions: Trigonometric Functions of a Negative Angle. The trigonometric functions of a negative angle may be converted into functions of a positive angle by use of the following formulas: Let the pupil supply the proofs for cot - x , sec - x , and csc - x. The same results are readily obtained for angles in the other quadrants by the use of appropriate diagrams. Reduction Tables and General Rules. Some of the reductions made by the methods of the preceding articles are used so frequently that it is convenient to collect the results obtained by them, and arrange them in tables for future reference.
But it must also be remembered that they are opposite in sign. But x meantime would be an angle in the first quadrant, hence sin x would be positive. In applying the above general rule to any particular example it will be found that the algebraic sign of the result is the same as the sign of the original function. If geometrical proofs for the above reduction formulas are, desired, such proofs may be obtained by following the methods of Art. This shows the pupil an important application of the preceding principle and introduces variety into the course of study.
But AT and BR are opposite in sign.
They are also B opposite in sign. By the general rule stated in Art. General Solutions of Trigonometric Equations. If there be no limit to the size of an angle, an indefinite number of angles will satisfy every trigonometric equation see Art. The pupil should observe that the values of x in a trigonometric equation differ in an important respect from the values of x in an algebraic equation.
Thus, in an algebraic equation the values of x are the roots of the equation and the number of values which x has equals the degree of the given equation. Whereas, for instance in Ex. A ship starting from a certain point sailed at the average rate of 9. At the end of 7 hr. If a railroad embankment is 11 ft. By dividing the triangle into right triangles and solving, find BC.
P is a spring of water, Q is a house, and R is a barn. If either x or y is obtuse, the above formulas may be proved as follows: In like manner the formula can be extended to the case where y is an obtuse angle. Hence, the above formulas are true when x and y are any angles. Formulas for sin x - y and cos x -y. By the same method as that used in Art. Prove the following identities: Functions of the Double Angle. By the formulas of Art. State the formulas for sin 2 x and cos 2 x in general language.
Find sin 3 x in terms of sin x. Find cos 3 x in terms of cos x. Find tan 3 x in terms of tan x. In a right triangle, C being the right angle, prove: Using the results of Exs. Also transform cos4 x into an expression in terms of cos 2 x and cos 4 x. Functions of the Half Angle.
State the formulas for sin 2 A, cos 1'A, and tan 1 A in general language. Express cos A, sin A, and cot A, in terms of cos 2 A. If the diagonal of a rectangle is in. Make up and solve a similar example for yourself. Adding and subtracting the formulas of Art. Reduce each of the following to a form adapted to logarithmic computation that is, to products or quotients: Compute the value of the expression in Ex.
Also of that of Ex. Make up for yourself an example similar to Ex. Besides those already arrived at, many other complex relations between the trigonometrical functions may be proved. Usually these relations are proved to the best advantage by reducing the two expressions, which are compared, to some common form, and hence inferring their identity by Ax. In most cases it is best to reduce given functions to sine and cosine. Prove that 1cos 2 A tan A. A - sin2 B. Functions of the Angles of a Triangle.
Find sin, cos, tan, cot, of: What nation first used the formula for sin I A? What man discovered the formula for sin 2 A? Who first published the formulas for sin A - B and cos A - B , and at what date? Law of Sines in a triangle. In any triangle the sides are to each other as the sines of the angles opposite. Let CD, denoted by p, be the altitude in each triangle. In like manner, b: Or, collecting results, a b c sinA sinB sin C Law of Tangents in a triangle.
In any triangle the szum of any two sides is to their difference as the tangent of half the sum of the angles opposite the given sides is to the tangent of haf the difference of these angles. In a triangle ABC Figs. This may be obtained as follows: Law of Cosines in a triangle. In the triangle ABC, Fig. But in the rt. If A is an obtuse angle, Fig. Formulas derived from the Cosine Formula. The formula for cos A in Art. In order to put this formula in such a shape that its value can be computed by the aid of logarithms, it is necessary to transform the numerator of the fraction into a product.
Prove that the diameter of a circle circumscribed about a triangle is equal to any side of the triangle divided by the sine of the angle opposite that side. By means of the property of sines, prove that the bisector of an angle of a triangle divides the opposite side into segments which are proportional to the sides forming the given angle. State this property in words.
Write the two similar formulas for b and c. Prove that the radius of an inscribed circle of a triangle is equal to c sin A sin 2 B where c is one side of the triangle and A and B cos I C are the angles adjacent to c, and C is the angle. What does the triangle become in each of these cases? When b sin B B is a right angle? Cases in the Solution of Oblique Triangles.
Four cases occur in the solution of oblique triangles according as the parts given are I. One side and two angles. Two sides and the included angle. Two sides and an angle opposite one of them. To solve Case I use the law of sines Art. The unknown sides may then be found by the following proportion: In solving oblique triangles by the use of logarithms it is of special importance to make an outline or skeleton of the work before looking up any logarithms, and then to do all the work connected with the use of the tables together.
Solve the oblique triangle ABC. Then by the law of sines Art. Any logarithm occurring more than once on being taken from the tables should be entered uniformly wherever it belongs. Proceeding thus, he should obtain log 2. In a parallelogram given a diagonal d, and the angles m and n which this diagonal makes with the sides, find the sides. Using four-place tables, find the unknown parts, having given: Without the use of tables, solve, having given: A ship S can be seen from two points M and N on the shore.
The distance MN is ft. Find the distance of the ship from M. A balloon is directly over a straight road, and between two points on the road from which it is observed. The distance between the two points is yd. Find the distance of the balloon from each of the given points, and also the height of the balloon from the ground. Which examples in Exercise 41 can be worked by Case I?
Work such of these examples as the teacher may direct. Make up some practical problem which can be solved by the method of Case I and solve it. To solve Case II we have the following method by the use of the law of tangents Art. The result will be half the sum of the unknown angles. One half of their difference may then be found by the following proportion: The third side is found by Case I. By the law of tangents Art. Using four-place tables, solve the following triangles, having given: Find C, B, and a. Find b without the use of tables. Solve the following, using either set of tables: Two trees M and P are on opposite sides of a pond.
The distance of 3M from a point K is Find the distance between the trees. Find the length of the lake. What will be the resultant force B and what angle will it make with each of -- the original forces? Use the principle 0 of the parallelogram of forces. If the trains travel at the rate of 40 and 50 miles an hour respectively, how far apart will they be in 10 minutes? The sides of a parallelogram are Find the two diagonals. In Exercise 41 which examples can be worked by the methods of Case II? Work such of these as the teacher may direct. Make up some practical problem which can be solved by the method of Case II and solve it.
In case it is desired to find only one of the angles of a given triangle it will be best to use that one of the formulas of Art. The cosine formula may be stated in general language thus: The cosine of one half of any angle of a triangle is equal to the square root of one half the sum of the three sides multiplied by one-half the sum minus the side opposite, divided by the product of the other two sides. In case the half angle T A to be computed is small, it is best not to use the formula for cos I A. In case it is desired to find all three angles of a triangle, it is best to use the tangent formula of Art.
For it will be found that by that method it is necessary to employ the logarithms of but four different numbers, whereas by either of the other formulas it is necessary to use the logarithms of seven different numbers. It is a further advantage to transform the tangent formula thus: Find the largest angle. The sides of a triangle are 10, 17, and Find the smallest angle in the triangle. The sides of a triangle are 3, 4, and 5. Find the sine of the smallest angle. The sides of a triangle are 1. Find the cosine of the largest angle. The sides of a triangle are 18, 21, and 25 ft.
Find the length of the perpendicular from the vertex of the largest angle to the opposite side. By use of four-place tables solve Exs. The distances between three towns, P, Q, R, are as follows: If R is due east from P, what is the direction of each place from every other place? What angle is subtended by an island 2 miles long as viewed from a point 3 miles distant from one end of the island and 4 miles from the other end?
But it has been shown in geometry that when two sides and an angle opposite one of them are given, sev- c eral special cases arise in the construction of the triangle. Then under the following conditions the following triangles may be constructed: If given z A is obtuse and 1. If given L A is right same results as in I. If given Z A is acute and 1. The case last mentioned 3 subdivides into three special cases as follows: In practice, the cases of no solution and of one right triangle or one isosceles triangle as the solution do not often occur.
Hence we usually need merely a method of discriminating between the cases where one oblique triangle or two. We may state this test in the form of question and answer thus: In general, when are there two solutions in Case IV? When the side opposite the given angle is less than the other given side. In this case, how may the two triangles be constructed? Take the vertex between the two given sides as a center, and describe an arc, using the smaller side as radius. Hence, in solving examples in Case IV, Observe whether the side opposite the given angle is less than the other given side; if it is, there are, in general, two solutions, which construct by taking the vertex between the given sides as a center and describing an arc with the smaller side as radius.
In case there are two solutions, use in one triangle the angle obtained from the table, and in the other triangle the supplement of this angle. Find the third angle and third side by Case I. In this case it is well to draw the smaller triangle separately as well as the general figure. By the law of sines Art. What checks can be used in the case of each of the two triangles? Using five-place tables, solve the following triangles, having given: In a parallelogram, one side is , one diagonal is Find the other side and other diagonal, and also the angles of the parallelogram.
Given two sides and the included angle, to find the area of a triangle, use the rule: The area of a triangle equals one half the product of any two sides multiplied by the sine of the angle included by these sides. For let the given sides be a and c. In case the given parts are a, b, C, or b, c, A, let the pupil state what the formula becomes. Let the pupil also state these formulas in general language. Given two angles and a side, find the third angle as usual. Let the given side be a, then a second side c may be determined as follows: In case two sides and an angle opposite one of them are given, to find the area it is necessary to find the log sin of the angle included between the two given sides by the method of Case IV Art.
In a parallelogram, given two adjacent sides, c and d, and the included angle A, obtain a formula for the area of the parallelogram in terms of the given parts. Prove that the area of any quadrilateral is equal to one half the product of its diagonals and the sine of their included angle. Find the area of the parallelogram without the use of tables. The diagonals of a quadrilateral are Find the area of the quadrilateral. Instruments for Measuring Angles. In order to determine unknown heights or distances it is important to have an instrument for measuring angles either in the horizontal or in the vertical plane.
Horizontal angles can be measured by the Surveyor's Compass. Both horizontal and vertical angles can be measured by the Transit Instrument. An angle of elevation is the angle between a line drawn from the eye of the observer to the point observed and the horizontal plane through the eye of the observer, when this angle is above the horizontal plane. An angle of depression is the angle between a line drawn from the eye of' the observer to the point observed and the horizontal plane through the eye of the observer, when this angle is below the horizontal plane.
To determine the Height of above a Horizontal Plane. In the right triangle ABC, what line shall we measure? How then can AB be computed? In the right triangle ABC, what side is known? What angle can be measured? How then can BC be computed? A Let AB, Fig. Or we may proceed by the use of natural tangents thus: To find the Distance of an Inaccessible Object.
Let the pupil determine what measurements and computations are necessary in accordance with the figure. To find the Distance between two Objects separated by an Impassable Barrier and possibly invisible to each other. Let it be required to find the disA, 7 tance between A and B Fig. Take a station C from which both A and 2B are visible.
In the triangle FIG. Let A and B Fig. In war, both on land and sea, the use of a range finder to determine the distance of an enemy is becoming general. The essential principle of such an instrument is the finding of the distance of an inaccessible object by the solution of a triangle in which a side called a base line and the two angles which include the side are known see Art. On land a convenient base line is taken and measured. In naval warfare, the distance between two points on the vessel is utilized as a base line. In the range finder the triangle employed is not usually solved by numerical computation, but by some mechanical method, which gives the result sought much more expeditiously.
Coast and Geodetic Survey. The essential parts of the work of the coast and geodetic survey are as follows:. The measurement of a base line AB Fig. The careful measurement of CD and the comparison of its computed length D, with the result of the measurement. By carrying these measurements far enough, a considerable arc of a great circle of the earth may be measured, and from this arc the radius or diameter of the earth computed. Distance of the Sun and Stars. The usual method of determining the distance of the sun from the earth consists essentially in taking a line AB, Fig.
The distance of the planet may then be computed by Art. The ratio of the distance of the sun to that of the planet from the earth being. The distance of the sun from the earth is thus found to be approximately 93,, miles. The distances of the fixed stars are found by taking the diameter of the earth's orbit as a base line, measuring the angles made by this line with lines drawn from its ends to a fixed star, and making the necessary computations.
Thus the trigonometrical solution of a triangle in which a side and the two angles adjacent to it are known is seen to have very wide practical applications. Trigonometry also has many applications to different departments of applied science.
This form of navigation is called Plane Sailing. The departure between two meridians is the arc of a parallel of latitude comprehended between the two meridians. Evidently the departure between two given meridians diminishes with the distance from the equator. The difference of longitude between two places is the angle at the pole or the arc on the equator included between the meridians of the two given places.
In Parallel Sailing a vessel sails due east or west i. In Middle Latitude Sailing a ship sails between two places in a course oblique to a parallel of latitude. For short distances especially near the equator sufficient accuracy is obtained by regarding the departure as measured on the parallel of latitude midway between the parallels of the two places, and computing the difference of longitude by the formula diff.
In Exercise 22 point out the examples which are solved by the method of Art. Also those which are solved by the method of Art. Also those solved by principles contained or implied in Art. The angle of elevation of the top of a tree measured from a point A water tower Find the distance of the observer from the base of the tower.
What is the altitude of the mountain above the Colorado plain? If the Colorado plain is ft. From the top of a hill ft. What is the distance from the foot of the tower to the top of the hill? Find the height of the tower. From the top and bottom of a castle which is 75 ft. Find the distance of the ship from the bottom of the castle. A monument 70 ft. Find the height of the tower and its distance from the monument. The three angles of a triangle are to each other as Find the other two sides.
Two mountains, A and B, are respectively 12 and 16 mi. Find the distance between the mountains. In a parallelogram one side is Find the other side and the other diagonal, also the angles of the parallelogram. A flagstaff 50 ft. Find the distance and height of the tower. The diagonals of a parallelogram are Find the sides of the parallelogram. The sides of a triangle are 11, 13, and From a point 4 mi. Find the length of the island. Two buoys are yd. Find the distance of the boat from the nearer buoy. Four miles from M on one road is the town of P, and 6 miles from M on the other road is the town of K.
How far apart are P and K? Find the distance TT. Two objects which are invisible from each other on account of a hill are visible from a station whose distances from the objects are yd. Find the distance between the objects. Given a circle with radius Find the area inclosed between two parallel chords on opposite sides of the center whose lengths are Find the distance from T to T'. Two yachts start at the same time from the same point, and sail one due west at the rate of 9.
How far apart will they be at the end of 2 hr. The following angles are then measured: Find the distance from the rock to the buoy. A ship sails due east mi. Find the difference in longitude which she makes. Find the difference of latitude and departure which she makes. Hence find her new latitude and longitude. A flagstaff 30 ft. Assuming the ground to be level, find the height of the building. A rock 60 ft. Find the distance between the observers, and the distance from the first observer to the base of the rock. A point at 0 is acted on B by a force which gives a velocity of ft.
If, in the figure of Ex. A tower ft. Find the distance between the boats. A man on the opposite side of a river from two trees P and Q wishes to determine the distance between the trees. He measures a distance A B, ft. A line is drawn so as to inclose, with the two paths, an acre of ground.
This line cuts one of the paths at a distance of 52 yd. What angle does this line make with each path? Find the length of the sides of the garden. Two towers are ft. The angle of elevation of one observed from the base of the other is twice that of the first observed from the base of the second; but from a point midway between the towers, the angles of elevation of the tops of the towers are complementary. Find the height of the towers. Do not use logarithms.
The length of the slope of the embankment on each side is 14 ft. Find the angle which the slope makes with the horizontal, and also find the width of the embankment at the base if the top is 8 ft. Find the distance DC. The area of a triangle is 3 acres and two of its sides are Find the angle between these sides. A shooting star is observed at two places mi. Taking the radius of the earth as mi. What is a shooting star? What causes its light? Show how to solve each of the cases in oblique triangles by dividing the oblique triangle into right triangles and using the methods of solving right triangles given in Chapter III.
Why do we not ordinarily use this method of solving oblique triangles? Make up or collect all the different examples you can showing practical applications of trigonometry, each example being distinct in principle or in field of application from the other examples. Radians, or the Circular Measure of Angles. The method of measuring angles by taking a right angle as the unit, dividing the right angle into 90 degrees, dividing each degree into 60 minutes, etc.
It continues to be generally used in spite of its awkwardness because of the extensive tables and large number of results stated in terms of it which have been accumulated. However, the advantages of the decimal division of any unit are so great that it is a growing custom to divide the degree of angle into tenths and hundredths instead of minutes and seconds see many examples in this book.
Also within the past century it has become customary in many kinds of work especially algebraic or theoretic work to use a unit of angle different from the right angle, called the radian, and to divide this unit decimally. The fundamental tools or instruments used in trigonometry are the functions of an angle now to be described and defined.
BC Then the ratio is termed the sine of the angle A. The sine of an acute angle is the ratio of the opposite leg to the hypotenuse. The cosine is the ratio of the adjacent leg to the hypotenuse. The tangent is the ratio of the opposite leg to the adjacent leg. The cotangent is the ratio of the adjacent leg to the opposite leg. The secant is the ratio of the hypotenuse to the adjacent leg. The cosecant is the ratio of the hypotenuse to the opposite leg. The versed sine is 1 minus the cosine. The coversed sine is 1 minus the sine.
These eight ratios are called the trigonometric ratios, or the trigonometric functions. The versed sine and the coversed sine are used so little in. Hence when we speak of the " six functions" we mean the first six trigonometric functions as given above. The abbreviations sin, cos, tan, cot, sec, csc, vers, covers, are ordinarily used for the eight functions.
The cosine, cotangent, cosecant, and coversed sine are termed the co-functions of the sine, tangent, secant, and versed sine respectively. In the above triangle Fig.
Utility of the Trigonometrical Ratios. These numerical values are used by what is virtually the geometrical principle of similar triangles in solving triangles. Later, however, they become units and elements which can be variously grouped and used in many kinds of algebraic processes. The value of a trigonometric function of an angle EB depends only on the size of the angle, not on the length of the lines chosen to B'v form the ratios.
Thus, by similar triangles in Fig. Given two sides of a right triangle, to compute the trigonometric functions for both acute angles of the triangle. In other words, he should make the small and capital letters as unlike as possible, and hence make them unlike in shape as well as in size. The reason for this is that the small and capital letters have entirely different meanings; and if as written by the pupil they have the same shape, the pupil is continually mistaking the small letter for the large, and vice versa.
Similarly the capital letter c should always be written in the form,g and not C. Write the functions of the acute angle B Fig. Let the teacher invert the triangle in various ways. Determine the value of the functions of A in the rt. A ABC, whose sides are a, b, c, if: Find the value of the functions of B in Exs. By the use of squared paper construct the angle whose Can you suggest some practical problem similar to that given in Art. What is the source of new power in trigonometry which enables us to do this? If by the methods of trigonometry we are able to solve any triangle in which one side and any two angles are given, suggest some practical problem which could be solved by this means and not by geometry.
By use of squared paper construct a rt. Show the same on Fig. Hence sin A, or -, is always less than 1. C What other function of A is always less than 1? Which functions of A are always greater than 1? Which may be either greater or less than 1? Which of the six functions are always proper fractions?
Verify this on Fig. If A is any acute angle, is it correct to say that secA is always greater than sin A? The values of which of the six functions of A on Fig. How many of the above examples can you work at sight i. Functions of the Complement of an Angle. Hence, in general, Any trigonometric function of an angle is equal to the cofunction of the complement of the angle. How many of the examples in this exercise can you work at sight? Three pairs of reciprocals exist among the trigonometric functions of an acute angle, viz.: Four equations connect the trigonometric functions of an acute angle in important ways.
Hence nine or more formulas give important values for the trigonometric functions. For from the results of Arts. One trigonometric function of an angle being given, the other functions may be found in either of two ways. By use of the formulas of Art. This consists of constructing a right triangle by use of the given function and deriving the required functions from the right triangle.
Construct a right triangle whose hypotenuse is 3 and altitude is 2, as ABC. Find by geometric methods squared paper may be used to advantage in constructing diagrams the other functions of A or x , given: Find by both methods the other functions of the angle named when: Express each of the other trigonometric functions of A in terms of: Which of the six functions are always less than 1?
Which are always greater than 1? How can you use this principle in testing the accuracy of examples like Exs. How many of the above examples can you work at sight? As stated in algebra, an identity is an equality which is true for all values of the unknown quantity or quantities contained in it. An equation proper or a conditional equation is an equality which is true only for a certain special value or values of the unknown quantity or quantities. Similarly in geometry the word " circle " is sometimes used to denote an area and sometimes a line the circumference , the context deciding in each case what is meant.
So 8" may mean either 8 inches or 8 seconds of angle, etc. Relations of identity among trigonometrical functions may be proved in either of two ways. By use of the formulas for the functions given in Arts. Instead of proving an identity by reducing one member of the identity to the form of the other, it is sometimes more advantageous to reduce both expressions to a common third form, and hence infer their identity by Ax.
By use of the values of the functions obtained by applying the definitions of the functions to the right triangle Art. Prove sin A cot A. In the solution of identities, the first of the two methods given above is to be preferred, since its use helps fix in mind the fundamental equations and formulas given in Arts. See if you can make up or discover any other trigonometrical identities for yourself.
It is helpful to notice that we determine these values in each case by the use of a right angle, the hypotenuse of which is taken as 1. Functions of and 6NO. Let AC be -BD. Of the results obtained in Arts. I 1 The results obtained in Arts. A OP in a quadrant whose radius is 1 is represented by the perpendicular let fall from P upon the radius OA. By methods which will be explained later see Art. These values are arranged in tables called Tables of Natural Trigonometric Functions.
By use of the table of natural tangents, construct: By use of the table of natural sines, construct: Find the numerical value of: Also when Z AOP equals Show that cos x is always less than cot x. If a flagstaff is at a distance of ft. Make up two examples similar to Ex. The Washington Monument is ft.
Find the distance of the monument from this place. At a certain spot ft. What is the perpendicular extent of the falls? Many trigonometric equations involving only acute angles may now be solved. If a church steeple is at a distance of 80 ft. If the height of the steeple is Make up an example similar to Ex.
In performing numerical work involving trigonometric functions, it is usually more expeditious to proceed by the use of logarithms. Hence the logarithms of the natural trigonometric functions have been obtained once for all and arranged in tables called Tables of Logarithmic Trigonometric Functions. The use of these tables is explained in the Introduction to the Tables Artso We have proved see Art.
By use of four-place tables, find A, given: Find the angle A if: Z log sin 9. L log cot 8. Z log tan Z log cos 8. Z log cos 9. Find angle A if: Z log cot 8.
Z log tan 8. Z log sin 8.
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Find the value of Find the vale of x tan Solve the following trigonometric equations: If at a distance of ft. Why are we able to determine this height by trigonometry and not by geometry? Who first, and at what date, defined the sine of an angle as the ratio between two lines see p. Give the different substitutes for this idea of the sine that had been used before this time. Why is the ratio definition of the sine superior to each of these? Explain the origin and literal meaning of the word sine see p.
Who first invented each of the other trigonometric ratios, and at what time see pp. Give some of the various names used for these ratios, with the names of the inventors of these names. Give an account of the computation of trigonometric tables see pp. Two Cases arise in the trigonometrical solution of right triangles.
Given one side and an acute angle. In each of these cases it will be observed that three parts are really given, since the right angle is known. The solution of Case I is effected as follows. This will give the unknown angle. The unknozon sides may then be found by means of the fbllowing: We first draw a diagram Fig. By use of four-place tables solve the right triangle in which b In the solutions of triangles fully one half the mistakes commonly made, and those the most important ones, are eliminated by making a rough mental forecast of the results before proceeding with the exact numerical work.
Thus in solving Ex. If then as a result of his exact numerical calculation, the pupil finds a leg greater than Similarly it is useful, by means of the rule and protractor, to make a drawing according to scale of the triangle to be solved, and from the figure to determine as accurately as possible the dimensions of the unknown parts by measuring them according to scale. Such results should be accurate enough to aid in eliminating any large errors in the numerical work. Indeed, if the work be neatly done, the results obtained from the diagram will be accurate enough for many practical purposes.
Exact checks of the numerical accuracy of the work of solving right triangles are obtained by calculating some side or angle of the triangle by a formula different from those already used in the computation, and observing whether the results thus obtained accord with those obtained in the first solution. Thus, to check the accuracy of the solution given for Ex. By use of five-place tables solve each of the following triangles, given: In working each example outline all the work carefully before looking up any logs-see Ex.
Find the remaining parts in each of the following right triangles, given: If the top of the Statue of Liberty in New York harbor is ft. Which of the examples in Exercise 22 are you able to solve by Case I? Solve one of these. Make up a similar practical problem for yourself and solve it, as for instance one concerning the Bunker Hill monument ft. Solve the following right triangles, by use of four-place tables, having given: Solve without the use of tables, having given: How many of Exs. The Solution of Case II is effected as follows: Find one of the angles of the given triangle by using that one of the following trigonometric ratios which contains the two given sides: Sources of Power in Trigonometrical Solution of Triangles.
There is danger that the pupil form mechanical habits of solving triangles without realizing the nature or. He should constantly realize that he is able to do what he is doing because some one before him has computed the legs of every possible right triangle whose hypotenuse is 1, and the other parts when each leg is 1, and arranged the results in tables natural sines, etc.
Also that some one else has made the pupil's work easier by looking up the logarithms of all the numbers in the natural tables and arranging them in other tables, and that the pupil is using this work also. Given the hypotenuse and a leg nearly equal, the angle between them will be very small. If this angle be found directly from the parts given, it will be found in terms of the cosine. Since the cosine of a small angle changes slowly as the angle varies, such a solution will not be accurate in the last figures.
B ijj a FIG. Washington railway at a certain place rises ft. The carpenter's rule for constructing 4 of a right angle is to construct a right triangle whose legs are 5 and 12 inches and take the greater acute angle in the triangle. How far is this from being correct?
The area of a right triangle may often be obtained more readily by the use of a formula involving only the particular parts of the triangle given. Product description Product Description Level Eighteen equips you with the numerous critical thinking skills you need to solve all types of trigonometric equations involving any angles including the special angles from Fab Five. By use of De Moivre's Theorem obtain the formulas for sin 3 x and cos 3 x. How many of the examples in this exercise can you work at sight? Usually these relations are proved to the best advantage by reducing the two expressions, which are compared, to some common form, and hence inferring their identity by Ax. In most cases it is best to reduce given functions to sine and cosine.
Which of the examples in Exercise 22 are you able to solve by the methods of Case II? Solve two of these. Make up a similar practical problem for yourself and solve it. Solve by use of four-place tables, having given: Without the use of tables solve in full each of the following right triangles, given: If certain parts of an isosceles triangle be given, the unknown parts may often be determined by dividing the isosceles triangle into two equal right triangles by means of a perpendicular drawn from the vertex to the base, and by solving one of the right triangles thus formed.
By use of four-place tables, solve the isosceles A triangle whose base is A regular polygon may be divided into equal right triangles by lines drawn from the center to the vertices and by the apothems to the sides. Hence if certain parts of a regular polygon are given, the remaining parts may often be determined by dividing the polygon into right triangles and R r solving one of these triangles. One side of a regular pentagon is Find the apothem, radius; perimeter, and area of the pentagon. One side of a regular decagon is 1. Find the apothem, radius, perimeter, and area of the decagon.
The radius of a circle is 16 feet. Find the side, apothem, and area of a regular inscribed dodecagon. Find the same magnitudes for a regular dodecagon which is circumscribed about a circle whose radius is The diagonal of a regular pentagon is 14; find the side, apothem, perimeter, and area of the pentagon. The apothem of a regular heptagon is 0. If mn denotes the base, h the altitude, I the leg, C the vertex angle, and D the base angle of an isosceles triangle, find: D, I, and C, in terms of nm and h.
D, C, and nm, in terms of h and 1. C, h, and 1, in terms of D and m. D, h, and I, in terms of C and nm. Solve the isosceles triangle in which a leg 2. If a chord If the radius of a circle is The side of a regular polygon of fourteen sides inscribed in a circle is Using four-place tables, solve the isosceles triangle in which: One side of a regular octagon is Find the apothem and area of the octagon.
The apothem of a regular pentagon is Find the perimeter of the pentagon. A regular decagon is inscribed in a circle whose radius is 1. Find the side and apothem of the decagon. Find the magnitude of the various parts of a regular heptagon circumscribed about a circle whose radius is The diagonal connecting two alternate vertices of a regular dodecagon is Find the side, apothem, and area of the dodecagon. If a chord of If the radius of a circle is , what is the length of a chord which subtends an arc of Find the base angle and altitude.
The leg of an isosceles triangle is , and the altitude is Find the base angle and base. The altitude of an isosceles triangle is 10, and the base angle is Find a leg and the base. The leg of an isosceles triangle is 6V2, and the base is Find the base angle, vertex angle, and altitude. The radius of a circle is 2. Find the number of degrees in an arc which subtends a chord whose length is 2V3. The diagonal of a square is Find the side of the square. General Method of computing Area of a Right Triangle. If b denote the.
To find log a and then the area we proceed as follows: Find the area of a right triangle in which the hypotenuse is and the base Formulas for Area of a Right Triangle. The area of a right triangle may often be obtained more readily by the use of a formula involving only the particular parts of the triangle given. By geometry, what is the method or formula for computing the area of an isosceles triangle? The formulas given above for computing the area of a right triangle are sometimes useful in computing the area of an isosceles triangle, or of a regular polygon. Compute the area of the isosceles triangle in which: Find the area of the regular pentagon whose perimeter is 3.
Find the area of the regular dodecagon whose apothem is 1. Find the area of a regular heptagon inscribed in a circle whose radius is 0. Given a regular octagon whose apothem is 2. If n denotes the number of sides, I? Solve the following right triangles, given: Find the area of each of the following isosceles triangles, given: In an isosceles triangle: Find the area of a regular decagon whose perimeter is Find the area of a regular pentagon whose apothem is.
Find the area of a regular heptagon inscribed in a circle whose radius is Given the side of a regular octagon as 5. Without the use of the tables, find the area of each of the following right triangles, given: Also of each of the following isosceles triangles, given: The angle of elevation see Art. How high is the cliff? At a point ft. What is the height of the tower? Find the height of the tree.
If the Eiffel Tower is 'ft. The length of a kite string is ft. Find the height of the kite supposing the kite string to be straight. One of the equal sides of an isosceles triangle is Find the base, altitude, and area of the triangle. What is the elevation 'of the sun, if a tree A ladder, 25 ft. Find the angle between the ladder and the house, and the distance the foot of the ladder is from the house. Why are we able to solve an example like this by trigonometry when we are not able to do so by geometry? If the Grand Canion of the Colorado is ft.
Find the width of the river. From the summit of a hill, there are observed two consecutive milestones on a straight horizontal road running from the base of the hill. The angles of depression see Art. Find the height of the hill. A valley is crossed by a horizontal bridge, whose length is 1. The sides of the valley make angles m and n with the plane of the horizon. Upon a hill overlooking the sea stands a tower 70 ft. What is the height of the hill and the horizontal distance of the ship from the tower?
Without the use of the tables find the length of all the other lines in the 0 figure. A boy standing m feet behind and opposite the middle of a football goal, sees that the angle of elevation of the nearer crossbar is A, and the angle of elevation of the crossbar at the other end of the field is C. Prove that the length of the field is m tan A cot C- 1. A railroad embankment is 7 ft. If the top of the embankment is 8 ft. Without the use of the tables find the height of the tree and the width of the river.
A tower and a monument stand on the same horizontal plane. The height of the tower is How high is the monument? A flagstaff stands on the roof of a building. From a point B, ft. Find the length of the flagstaff. From the top of a lighthouse, ft. Find the distance, in feet, of the buoy from the shore. The base of a rectangle is Find the altitude of the rectangle and the angle which the diagonal makes with the base. The Singer building of New York City is ft. Make up some problem concerning this which can be solved by trigonometry. The diagonals of a rhombus are Find the sides and angles.
Make up or collect as many different examples as you can showing the practical uses of the solution of right triangles by trigonometry, each example being distinct from the rest either in principle or in the field of its application. Who first, and at what date, taught the trigonometric solution of triangles in the same general way as is done at present? Hence it is important to learn what the trigonometric functions of an obtuse angle are.
In astronomy, the heavenly bodies, by successive rotations about an axis, and by revolutions in an orbit, also generate angles unlimited in size.
Hence a general method is needed of determining the trigonometric functions of angles unlimited in size. In treating of the properties of angles in general, it is convenient, wherever possible, to let the angles start at the same place, as OA that is, to have the vertex and a side in common.
Let the rotating radius start in the position OA and rotate toward the position OB in the direction contrary to'that in which the hands of a clock move, or counter-clockwise. The initial line of an angle is the rotating radius, which generates the angle, in its first position, as A O. In general an angle is said to be of or in that quadrant in which its terminal line lies.
In algebra it is shown that negative quantity is quantity exactly opposite in some respect, as, for instance, in direction, from other quantity taken as positive. Hence if the rotating radius move from the position OA Fig. If the radius continue to rotate in this direction, a whole series of negative angles will be formed similarly.
The Y signs of other lines used are deFIG. The origin is the point in which the axes intersect, as the point 0 on Fig. The ordinate of a point is the distance of the point above or below the axis XX'. The abscissa of a point is the distance of the point to the right or left of the YY' axis. Coordinates is the general term for abscissa and ordinate of a point. The coordinates of a point may be written together in parenthesis with abscissa first and a comma between.
The distance of a point is the line drawn from the origin to the point, thus on Fig. The distance of a point is independent of sign. Definitions of Trigonometric Functions of Any Angle. Trigonometric Functions represented by Lines. If a circle Fig. Or, in the circle as described, the sine of an angle is represented by a line drawn from the terminal end of the arc intercepted by the angle, and perpendicular to the horizontal diameter.
Or, in the circle as described, the cosine of an angle is represented by a line drawn from the terminal end of the arc intercepted by the angle, and perpendicular to the vertical diameter. Or in the circle as described, the tangent of an angle is represented by a line drawn touching the initial end of the arc intercepted by the angle, and terminated by the radius to the other end of the arc, produced. It will be convenient to draw a figure for an angle in each quadrant showing the lines which represent the functions of that angle.
The lines which represent the various trigonometric functions of an angle are not the same as the trigonometric functions which they represent, but they have many of the same properties as the functions or ratios. Signs of the Trigonometric Functions in the Different Quadrants. Of the lines representing the sines of angles in the different quadrants, viz.
The students may obtain the same results from Figs. Since the sine of a quantity and of its reciprocal must be the same, the sign of the cotangent in the various quadrants must be the same as that of the tangent; that of the secant, the same as the cosine; that of the cosecant, the same as the sine. Or, proceeding geometrically, on Fig. The secant is considered as plus when it is drawn in the same direction from the center as the terminal radius thus OT2, Fig.
Similarly the cosecant lines Fig. The results thus obtained may be arranged in a table as follows: Find the signs of the functions of the angles in Exs. Give two positive and two negative angles each of which is coterminal with: Find the smallest possible angle coterminal wvith: In which quadrant does an angle lie: If its sin is positive and cos negative?
If its tan is positive and sin negative? If its cot is negative and cos negative? If its csc is negative and cot positive? If its cos is positive and tan negative? If its sec is negative and tan negative? A railroad embankment is 9 ft. If a railroad embankment is 7 ft. Make up a similar example for yourself. Similar results are readily perceived for other quadrants by the use of a figure showing the sines as lines in the different quadrants.
Similarly in the first quadrant Fig. Formulas for the Acute Angle extended to any Angle. The equations and formulas proved in Arts. Thus, on each of the Figs.