Write the expression as an algebraic expression in x for x> 0

Question

Write the expression as an algebraic expression in x for x > 0

sin[2sin-1 x/4]

Answer 1

sin(2sin^-1[x/4])

sin^-1[x/4] is an angle

call the angle θ

sinθ=x/4

draw a right triangle and find the cosine

sin=opposite/hypotenuse

cos=adjacent/hypotenuse

call the adjacent side "y"

use the Pythagorean Theorem

the hypotenuse is 4 and one leg is x and the other leg is "y"

4^2=x^2+y^2

y^2=16-x^2

y=√(16-x^2)

now find the cosine

cosθ=√(16-x^2)/4

sin 2θ=2sinθcosθ

2sinθcosθ=2(x/4)(√(16-x^2)/4)

2sinθcosθ=(x/2)(√(16-x^2)/4)

2sinθcosθ=x√(16-x^2)/8 is the expression in terms of x