Chapter 8, Problem 1RE

To determine

Whether the statement “If $\theta$ is a fourth quadrant angle, then $\cos \theta >0$” is true or false.

Answer to Problem 1RE

The given statement is $\underset{\_}{\text{true}}$.

Explanation of Solution

It is a known fact that all trigonometric ratios are positive in the first quadrant $\left(0°\text{ to 90}°\right)$.

The trigonometric ratios sine and its reciprocal cosecant are positive in the second quadrant $\left(\text{90}°\text{ to 180}°\right)$.

The trigonometric ratios tangent and its reciprocal cotangent are positive in the third quadrant $\left(\text{180}°\text{ to 270}°\right)$.

The trigonometric ratios cosine and its reciprocal secant are positive in the fourth quadrant $\left(\text{270}°\text{ to 360}°\right)$.

Represent the above facts more clearly as shown in below Figure 1.

From Figure 1, it can be concluded that if $\theta$ lies in fourth quadrant, then $\cos \theta$ is positive.

Therefore, the given statement “If $\theta$ is a fourth quadrant angle, then $\cos \theta >0$” is $\underset{\_}{\text{true}}$.