Chapter 4.7, Problem 68E

Solution

(a)

To calculate: Decompose each of the following algebraic function.

  $\cos (\arcsin (x))$

Given information: $\sqrt{1-x^2}$

Calculation: first, from a triangle with side length that will allow you to compute the given expression and $x$ itself using trigonometric values of one of the angles.

Then substitute one of these values into the other to form the trigonometric decomposition.

  $\cos (\arcsin (x))$

(b)

To calculate: Decompose each of the following algebraic function.

  $\sin (\arctan (x))$

Given information: $\frac{x}{\sqrt{1+x^2}}$

Calculation: first, from a triangle with side length that will allow you to compute the given expression and $x$ itself using trigonometric values of one of the angles.

Then substitute one of these values into the other to form the trigonometric decomposition.

  $\sin (\arctan (x))$

(c)

To calculate: Decompose each of the following algebraic function.

  $\cot (\arccos (x))$

Given information: $\frac{x}{\sqrt{1-x^2}}$

Calculation: first, from a triangle with side length that will allow you to compute the given expression and $x$ itself using trigonometric values of one of the angles.

Then substitute one of these values into the other to form the trigonometric decomposition.

  $\cot (\arccos (x))$