Chapter 7.4, Problem 5AYU

$\cos (-\theta )-\cos \theta =$ _____.

To determine

Value of trigonometric identity in the form $\text{cos}\left(-\theta \right)-\text{cos}\theta$.

Answer to Problem 5AYU

Solution:

Value of trigonometric identity in the form $\text{cos}\left(-\theta \right)-\text{cos}\theta =0$.

Explanation of Solution

Given:

Trigonometric identity is in the form of $\text{cos}\left(-\theta \right)-\text{cos}\theta$.

Calculation:

Consider a unit circle.

Equation of a unit circle $=x^2+y^2=1$   -----1

If $P\left(x,y\right)$ is a point on the unit circle that corresponds to angle $\theta$ then

$x=\text{ cos}\theta$   -----2

Using the symmetry, the point $Q$ on the unit circle corresponds to the angle $\left(-\theta \right)$.

The point $Q$ has the coordinates $\left(x,-y\right)$.

Using values from equation 2 to obtain the coordinates of $Q$.

$Q(x,-y)=Q\left(\text{cos}\theta ,-\text{sin}\theta \right)$   -----3

By definition of trigonometric function

$\text{Cos}\left(-\theta \right)=x=\text{ cos}\theta$   -----4

$\text{Cos}$ is an even function.

From equation 4

Value of trigonometric identity in the form $\text{cos}\left(-\theta \right)-\text{cos}\theta =0$.