Chapter 6.3, Problem 36E

To determine

To calculate: The quadrant in which the angle $\theta$ lies if $\tan \theta <0$ and $\sin \theta <0$ .

Answer to Problem 36E

The quadrant in which the angle $\theta$ lies is $\text{IV quadrant}$ .

Explanation of Solution

Given information:

The sign of trigonometric functions $\tan \theta <0$ and $\sin \theta <0$ .

Formula used:

Coordinate plane is divided into four quadrants.

In the first quadrant all trigonometric functions that is $\sin \theta ,cos\theta ,\tan \theta ,\csc \theta ,\sec \theta \text{ and }\cot \theta$ are positive.

In the second quadrant only sine and cosecant trigonometric functions that is $\sin \theta \text{ and }\csc \theta$ are positive and remaining $cos\theta ,\tan \theta ,\sec \theta \text{ and }\cot \theta$ are negative.

In the third quadrant only tangent and cotangent trigonometric functions that is $\tan \theta \text{ and }\cot \theta$ are positive and remaining $cos\theta ,sin\theta ,\sec \theta \text{ and }\csc \theta$ are negative.

In the fourth quadrant only cosine and secant trigonometric functions that is $\cos \theta \text{ and }\sec \theta$ are positive and remaining $\sin \theta ,\tan \theta ,\csc \theta \text{ and }\cot \theta$ are negative.

Calculation:

Consider the provided sign of trigonometric functions $\tan \theta <0$ and $\sin \theta <0$ .

Recall that coordinate plane is divided into four quadrants.

In the fourth quadrant only cosine and secant trigonometric functions that is $\cos \theta \text{ and }\sec \theta$ are positive and remaining $\sin \theta ,\tan \theta ,\csc \theta \text{ and }\cot \theta$ are negative.

According to the condition both sine and tangent trigonometric functions are negative which is possible only in fourth quadrant.

Thus, the quadrant in which the angle $\theta$ lies is $\text{IV quadrant}$ .