Chapter 7, Problem 5RCC

(a)

To determine

To state: Double-Angle Formulas for Sine, Cosine and Tangent.

Answer to Problem 5RCC

The Double-Angle formulas for Sine, Cosine and Tangent are explained below.

Explanation of Solution

Given information:

The trigonometry functions are sine, cosine and tangent.

Calculation:

The Double-Angle formula for sine is as follows.

  $\sin 2x=2\sin x\cos x$

The Double-Angle formulas for cosine are as follows.

  $\begin{array}{c}\cos 2x={\cos }^{2}x-{\sin }^{2}x\\ \cos 2x=1-2{\sin }^{2}x\\ \cos 2x=2{\cos }^{2}x-1\end{array}$

The Double-Angle formula for tangent is as follows.

  $\tan 2x=\frac{2\tan x}{1-{\tan }^{2}x}$

(b)

To determine

To state: The formulas for lowering powers.

Answer to Problem 5RCC

The formulas for lowering powers are explained below.

Explanation of Solution

Given information:

The trigonometry functions are sine, cosine and tangent.

Calculation:

The formula for lowering powers of sine is as follows.

  ${\sin }^{2}x=\frac{1-\cos 2x}{2}$

The formula for lowering powers of cosine is as follows.

  ${\cos }^{2}x=\frac{1+\cos 2x}{2}$

The formula for lowering powers of tangent is as follows.

  ${\tan }^{2}x=\frac{1-\cos 2x}{1+\cos 2x}$

(c)

To determine

To state: The Half-Angle formulas.

Answer to Problem 5RCC

The Half-Angle formulas are explained below.

Explanation of Solution

Given information:

The trigonometry functions are sine, cosine and tangent.

Calculation:

The Half-Angle formula for sine is as follows.

  $\sin \frac{u}{2}=\pm \sqrt{\frac{1-\cos u}{2}}$

The Half-Angle formula for cosine is as follows.

  $\cos \frac{u}{2}=\pm \sqrt{\frac{1+\cos u}{2}}$

The Half-Angle formula for tangent is as follows.

  $\begin{array}{c}\tan \frac{u}{2}=\frac{1-\cos u}{\sin u}\\ =\frac{\sin u}{1+\cos u}\end{array}$

The choice of + or − sign depends on the quadrant in which u /2 lies.