Inverse trig functions?

Question

Find the exact value without a calculator

sin^ -1[cos( π/4)]

cos[2 sin^ -1 (1/2)]

Answer 1

These essentially are asking you to find which quadrant the resulting angle ends up.

First problem: π/4 is in quadrant 1. The sides of the triangle representing this angle are sqrt(2)/2 each, with the hypotenuse being 1. So cos(π/4) = [sqrt(2)/2]/1=sqrt(2)/2. Now, we need to know which angle causes the inverse sine to be sqrt(2)/2. This answer, coincidentally, is also π/4.

Second problem: sine inverse of 1/2 is asking which angle gives a sine of 1/2 as a result (that is 30-degrees and 150-degrees, +/- 360*n degrees). However, the sine inverse function is only defined in quadrant 1 and quadrant 4 (without the +/- 360*n degrees). Thus the only angle that fits in those allowed quadrants is the 30-degree answer. So, 2*30 = 60 (since you had [2 sin^ -1 (1/2)], we had to double the angle). Now, cos(60) = 1/2 (coincidentally the same as the argument of the sine inverse function).

Answer 2

The first one is the angle whose sin is cos 45° which is 1/√2, which is 45°

The second one is the cos of the angle whose sin is 1/2. The angle whose sin is 1/2 is 30°. The cos of 30° is √3/2.

Answer 3

sin^ -1[cos( π/4)]
= sin^ -1[sqrt(2)/2]
=π/4

cos[2 sin^ -1 (1/2)]
=cos[2 x π/6]
=cos(π/3)
=1/2

Answer 4

assuming first quadrant:

sin^-1 [cos(pi/4)]
sin^-1[1/(sqrt 2)]
pi/4

cos[2 sin^-1 (1/2)]
cos[2 (pi/6)]
cos[pi/3]
1/2

Answer 5

sin^-1(sqrt2/2)= pi/4

cos(2(pi/6)..cos(pi/3)= 1/2