Evaluate without a calculator. A.) cos−1 (sin 5𝜋/3) B.) sin−1 (− √3/ 2 ) C.) sin−1(tan(−2))

Answer 1

Evaluate without a calculator. A.) cos−1 (sin 5𝜋/3) B.) sin−1 (− √3/ 2 ) C.) sin−1(tan(−2))

okay

A) is same as cos^-1(cos(𝜋//2- 5𝜋/3) since cos( (𝜋/2-5𝜋/3)=sin(5𝜋/3) so

answer is angle = -2𝜋/3;;;; answer for cos^-1(value) is an angle

B) is same as sin^-1sin(-𝜋/3) answer is angle -𝜋/3

C)tan(-2) describes tan of angle -2 radiians. which I believe has value less than -1, and sin^-1 domain is llimited to only -1 to 1. I think there is no angle.

Answer 2

a) cos^-1(sin (5π/3)) = cos^-1(sin(-π/3)) = cos^-1(-sin(π/3)) = cos^-1(√3/2). This can be either 5π/6 or 7π/6, if you look at your unit circle. Just find the values x where cos x = √3 / 2

b) sin^-1((√3)/2). Just look at what values of x give sin x = √3 / 2. In this case x = π/3 and x = 2π/3.

c) sin^-1(tan(-2)). This cannot be evaluated as written, since tan(-2) is larger than 1 (since -3π/4< -2 <- π/2), and sin^-1(x) only works for |x|≤1. It is undefined.