The values of the last two angles in radians are obtained from page 32 of the Tables. It is essential that these results be checked, as extraneous roots are frequently introduced when an equation is squared. We have: The first and last values of x do not satisfy the given equation, and they are therefore discarded. The values = π and x = 0.9272 do satisfy the equation. All other solutions may be obtained from these by adding 2nπ. 119. Trigonometric Identities. Identities constitute an important part of mathematics. If we encounter a complicated fraction in the solution of a problem, we proceed to simplify it. The reduction of the fraction is essentially a proof of the identity obtained by equating the original fraction to the simpler result which is found. which is proved by Theorems 7 and 8, page 223, is the basis of computing b, a side of a triangle, if a, A, and B are given. Trigonometric identities are of the same importance in dealing with expressions involving the trigonometric functions as algebraic and logarithmic identities are in working with algebraic and logarithmic functions. Trigonometric identities involving functions of a single angle may be proved by means of the fundamental formulas in Section 117 and the operations of algebra, by transforming one member of the identity into the other. The proofs of equations (6) and (7) in the section cited are illustrations of the method. Another method of establishing the truth of an identity is to show that both members can be reduced to the same form. Still another is to transform the given equation until it is reduced to some known identity. EXERCISES 1. Express formulas (1)–(8), Section 117, in words. 2. By means of the formulas in Section 117, find all the functions of 0, given = (b) tan 0 = 3. (c) csc 0 = 3. (a) cos 0 3. Using both methods in the Example in Section 117, express each of the functions in terms of (a) cos 0. (b) tan 0. (c) cot 0. (d) sec 0. (e) csc 0. 120. Functions of the Sum of Two Angles. The relation (a + b)2 = a2+2ab+b2 expresses in a different form the square of the sum of two numbers. We seek now an analogous expression for the sine of the sum of two angles, sin (0 + ), where 0 and are any two acute angles. The sum + is constructed by using the terminal line of 0 as the initial line of p. It may be either an acute or an obtuse angle, as indicated in the figures. Let A be any point on the terminal line of 0, and draw AB perpendicular to OX. Then To express this ratio in terms of 0 and 4, draw AC perpendicular to the initial line of p, and CD perpendicular to OX. Then the functions of 0 and may be found from the right triangles ODC and OCA respectively. Draw CE perpendicular to AB. Then the triangles ODC and AEC are similar (why?), so that EAC = 0, and the functions of may also be found from the triangle AEC. We then have The proofs hold for either figure. In deriving (2) when 0+ is obtuse, it should be noted that OB is negative, so that we will still have OB OD - BD. = We shall assume that these formulas hold for all values of and, negative as well as positive. (See Exercises 5-8 below.) Dividing (1) by (2), and using (4), page 333, we have Dividing numerator and denominator by cos e cos, we 1. Express formulas (1), (2), (3), in words. 2. Using the functions of 30°, 45°, 60°, as given on page 161, find all the functions of (a) 75°, (b) 105°. Hint: sin 75° = sin (45° + 30°), etc. 3. If 0 arc sin and = arc cos, find the functions of 0 + . 4. What force acting parallel to the plane is necessary to support a body weighing 100 pounds on a smooth plane if the inclination is = arc sin? If the inclination is arc sin ? If the inclination is 0 + ¢? = 5. Prove that (1) holds for all positive values of 0 and 4. Solution: We will first prove that: If (1) and (2) are true for a pair and ☀, then (1) is true when ✪ is increased by 90°. of values Since (1) holds for all acute values of 0 and 4, by the above it holds for all obtuse values of ℗ and all acute values of 4. |