Pre-calculus math

Question

Find the exact value of the expression. Sin(cos^-1(3/4)-tan^-1(1/3))

Answer 1

Answer 2

Can you see that this requires you to use the angle subtraction identity?

sin(x – y) = sin(x)cos(y) – cos(x)sin(y)

where x = cos^-1(3/4) and y = tan^-1(1/3)

So we would break the expression up into 4 parts:

1) sin(cos^-1(3/4))

2) cos(tan^-1(1/3))

3) cos(cos^-1(3/4))

4) sin(tan^-1(1/3))

Since the inverse trig functions would all refer to Q1 for positive values, we can draw triangles in Q1 to determine exact values:

For sin(cos^-1(3/4) because cos(θ) = adjacent/hypotenuse, the triangle looks like this:

Calculate the 3rd side using the Pythagorean Theorem: 3² + x² = 4² or x = √7 therefore, sin(θ) = √7/4 so the expression sin(cos^-1(3/4)) = √7/4

Repeat this process for each of the other 3 pieces then multiply/subtract using the angle subtraction identity.

As a clue, the second triangle would look like this: