Chapter C, Problem 19E

(a)

To determine

To prove : $\sin \left(-\theta \right)=-\sin \left(\theta \right)$

Explanation of Solution

Given information :

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

Concept used :

There are 4 quadrants in all and we know that in 1st quadrant , all 6 trigonometric functions $i.e\sin \theta ,\cos \theta ,\sec \theta ,\csc \theta ,\tan \theta$ and $\cot \theta$ take positive values.

In 2nd quadrant , only $\sin \theta$ and $\csc$ functions take positive values and other left 4 functions take negative values .

Similarly, in 3rd quadrant , $\tan \theta$ and $\cot \theta$ functions take positive values and left 4 functions take negative values.

And in 4th quadrant , $\cos \theta$ and $\sec \theta$ functions will take positive values and left ones will take negative values .

Proof:

$\sin \left(-\theta \right)$ can be written as $\sin \left(0-\theta \right)$

Now, referring to 4 quadrants we have in geometry, we will see that $\left(0-\theta \right)$ lies in 4^th quadrant

And since only cos and sec functions take positive values in 4^th quadrant

$\Rightarrow \sin \left(0-\theta \right)\text{ or sin}\left(-\theta \right)$ comes out as $-\sin \left(\theta \right)$ because sin function takes negative value in 4^th quadrant.

Hence, proved

(b)

To determine

To prove : $\cos \left(-\theta \right)=\cos \left(\theta \right)$

Explanation of Solution

Given information :

  $\cos \left(-\theta \right)=\cos \left(\theta \right)$

Concept used :

There are 4 quadrants in all and we know that in 1st quadrant , all 6 trigonometric functions $i.e\sin \theta ,\cos \theta ,\sec \theta ,\csc \theta ,\tan \theta$ and $\cot \theta$ take positive values.

In 2nd quadrant , only $\sin \theta$ and $\csc$ functions take positive values and other left 4 functions take negative values .

Similarly, in 3rd quadrant , $\tan \theta$ and $\cot \theta$ functions take positive values and left 4 functions take negative values.

And in 4th quadrant , $\cos \theta$ and $\sec \theta$ functions will take positive values and left ones will take negative values .

Proof:

$\cos \left(-\theta \right)$ can be written as $\cos \left(0-\theta \right)$

Now, referring to 4 quadrants we have in geometry, we will see that $\left(0-\theta \right)$ lies in 4^th quadrant

And since only cos and sec functions take positive values in 4^th quadrant

$\Rightarrow \cos \left(0-\theta \right)\text{ or cos}\left(-\theta \right)$ comes out as $\cos \left(\theta \right)$ because cos function takes positive value in 4^th quadrant.

Hence, proved