Chapter 6.3, Problem 38E

To determine

To calculate: The quadrant in which the angle $\theta$ lies if $\csc \theta >0$ and $\cos \theta <0$ .

Answer to Problem 38E

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

Explanation of Solution

Given information:

The sign of trigonometric functions $\csc \theta >0$ and $\cos \theta <0$ .

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 $\csc \theta >0$ and $\cos \theta <0$ .

Recall that coordinate plane is divided into four quadrants.

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.

According to the condition cosecant trigonometric function is positive and cosine trigonometric function is negative which is possible only in second quadrant.

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