Chapter 7.4, Problem 6AYU

True or False $\sin \left(-\theta \right)+\sin \theta =0$ for any value of $\theta$.

To determine

Value of trigonometric identity $\text{sin}\left(-\theta \right)+\text{sin}\theta =0$ for any value of $\theta$.

Answer to Problem 6AYU

Solution:

True. Value of trigonometric identity $\text{sin}\left(-\theta \right)+\text{sin}\theta =0$ for any value of $\theta$.

Explanation of Solution

Given:

Trigonometric identity is in the form of $\text{sin}\left(-\theta \right)+\text{sin}\theta =0$.

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

$y=\text{ sin}\theta$   -----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

$\text{sin}\left(-\theta \right)=-y=-\text{sin}\theta$   -----4

$\text{sine}$ is an odd function.

From equation 4

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