Question

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

Short answer

Given statement is true statement

Step-by-step solution

Given information

We are given an trigonometric identity$\sin \left(-\theta \right)+\sin \left(\theta \right)$

Explanation for true statement

We know that sine is an odd function

Hence $\sin \left(-\theta \right)=-\sin \left(\theta \right)$

Hence we get,

$$\begin{gathered} \sin \left(-\theta \right)+\sin \left(\theta \right) \\ -\sin \left(\theta \right)+\sin \left(\theta \right) \\ =0 \end{gathered}$$

For any value of theta

Explanation for false statement

Suppose

$\sin \left(-\theta \right)+\sin \left(\theta \right)\ne 0$

But sine is an odd function

hence $\sin \left(-\theta \right)=-\sin \left(\theta \right)$

Which is a contradiction to our assumption

Hence our assumption is wrong

And therefore

$$\begin{gathered} \sin \left(-\theta \right)+\sin \left(\theta \right) \\ -\sin \left(\theta \right)+\sin \left(\theta \right) \\ =0 \end{gathered}$$

Conclusion

Given statement is true statement