Trig problem

Question

Evaluate sin(sin^-1(sin60)) [60 degrees]

 

and 

 

evaluate cos^-1(cos(cos^-10.5)) if the angle is in quadrant 1. 

Answer 1

 

 

sin(angle) = opposite/hypotenuse

 

sin^-1(opposite/hypotenuse) = angle

 

Evaluate:

 

sin(sin^-1(sin(60))) 

 

We know: sin(60º) = √(3)/2

so the inverse sine function, sin^-1(√(3)/2) = 60º

 

Then;

 

sin(sin^-1(sin(60º))) 

 

sin(sin^-1(√(3)/2))

 

sin(60º)

 

= √(3)/2

 

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Evaluate:

 

cos^-1(cos(cos^-1 0.5)) if the angle is in quadrant 1. 

 

We know: cos(60º) = 1/2

so then the inverse cosine function; cos^-1(0.5) = 60º

 

cos^-1(cos(cos^-1 0.5))

 

cos^-1(cos(60º))

 

cos^-1(0.5)

 

= 60º

 

Answer 2

Evaluate sin(arcsin(sin60^o))

 

 

Let's simplify, working from the inside out.  To begin, sin^-1 (sin60^o) = 60^o because the inverse cancels sin.

 

Now this has simplified to sin(60^o) = ½•√3.

 

Similar work will help you solve the second problem.