Using a reference triangle, find the exact value of the expression: (Also is there any difference between using a reference triangle to find the exact value...

Question

...or to just find the exact value)

 

cos (sin⁻¹ x)

 

a. √(1+x²)

b. √(1-x²)

c. x/(√1-x²)

d. x/(√1+x²)

Answer 1

Let θ = sin^-1x   So, sinθ = x = x/1

 

Draw a right triangle (reference triangle) with an angle θ, the side opposite θ of length x, and hypotenuse with length 1.  Then, by the Pythagorean Theorem, the side adjacent to θ has length √(1-x²).

 

So, cos[sin^-1x] = cosθ = (side adjacent to θ)/hypotenuse 

 

                               = √(1-x²)/1 = √(1-x²)

 

ANSWER: (b)

 

NOTE:  In the above, it is assumed that x > 0.  If x < 0, then the opposite side has length -x and by the Pythagorean Theorem, the adjacent side has length √(1 - (-x)²) = √(1-x²).  We still get answer (b) for cos[sin^-1x] if x < 0.