The above results are also the sine and cosine, respec tively, of 54° (Art. 15). Otherwise thus: Let x=36°; then 2x=72°, and 3x=108°. But 36° is an acute angle, and therefore its cosine is positive. 59. If A+B+C = 180°, or if A, B, C are the Angles of a Triangle, prove the Following Identities: A B C (1) sin A + sin B + sin C = 4 cos COS COS 2 A (2) cos A+ cos B + cos C = 1+4 sin sin (3) tan Atan B + tan C tan A tan B tan C. B C 2 (3) .. tan Atan B+tan C = tan A tan B tan C NOTE. The student will observe that (1), (2), and (3) follow directly from Examples 1 and 2, and formula (3), respectively, of Art. 48, by putting A+B+C= 180°. EXAMPLES. Prove the following statements if A+B+C= 180°: 1. cos (A + B — C) = cos 2 C. 3. sin 2 A+ sin 2 B + sin 2C4 sin A sin B sin C. 4. sin 2 A+ sin 2 B - sin 2C = 4 sin C cos A cos B. The equation 60. Inverse Trigonometric Functions. sin x means that is the angle whose sine is x; this may be written sin-1x, where sin-1x is an abbreviation for the angle (or arc) whose sine is x. = So the symbols cos-1x, tan-1x, and sec-1y, are read "the angle (or arc) whose cosine is x," "the angle (or arc) whose tangent is a," and "the angle (or arc) whose secant is y." These angles are spoken of as being the inverse sine of x, the inverse cosine of x, the inverse tangent of x, and the inverse secant of y, respectively. Such expressions are called inverse trigonometric functions. NOTE. -The student must be careful to notice that -1 is not an exponent, sin1x is not (sin x)-1, which 1 sin x Notice also that sin-13-cos-11 is not an identity, but is true only for the par ticular angle 60°. 2 2 This notation is only analogous to the use of exponents in multiplication, where we have a-1a=a0=1. Thus, cos1 (cos x) = x, and sin (sin x) = x; that is, cos-1 is inverse to cos, and applied to it annuls it; and so for other functions. The French method of writing inverse functions is arc sinx, arc cosx, arc tan x, and so on. EXAMPLES. 1. Show that 30° is one value of sin1. We know that sin 30° = . .. 30° is an angle whose sine is ; or 30° sin1. 2. Prove that tan-1+tan-1} = 45°. tan-1 is one of the angles whose tangent is, and tan-1 is one of the angles whose tangent is Therefore 45° is one value of tan-1 + tan-1. Any relations which have been established among the trigonometric functions may be expressed by means of the inverse notation. Thus, we know that Put cos 0=x. .. 2 cos-1x cos-1(2x2-1). 6. By Art. 49, sin 20 = 2 sin cos 0, which may be written 20 sin (2 sin cos 0). = Put sin0= x. .. 2sin-1x sin−1(2 x √1 − x2). = |