Chapter 12, Problem 3CR

To determine

Find the product of all six trigonometric functions of an angle $\left(\theta \right)$ in standard position whose terminal side passes through $\left(-\sqrt{3},1\right)$ .

Answer to Problem 3CR

The value of product of all six trigonometric functions is 1. Therefore option (b) is correct.

Explanation of Solution

Given information: The terminal side of an angle $\left(\theta \right)$ in standard position passes through $\left(-\sqrt{3},1\right)$ .

Calculation: Since for any angle $\left(\theta \right)$ ,

  $\begin{array}{c}\sin \theta =\frac{1}{\csc \theta }\\ \cos \theta =\frac{1}{\sec \theta }\\ \tan \theta =\frac{1}{\cot \theta }\end{array})\mathrm{....}\left(1\right)$

Re-write (1).

  $\begin{array}{c}\sin \theta \cdot \csc \theta =1\mathrm{....}\left(2\right)\\ \cos \theta \cdot \sec \theta =1\mathrm{....}\left(3\right)\\ \tan \theta \cdot \cot \theta =1\mathrm{....}\left(4\right)\end{array}$

Multiply (2), (3) and (4) gives,

  $\sin \theta \cdot \csc \theta \cdot \cos \theta \cdot \sec \theta \cdot \tan \theta \cdot \cot \theta =1$

So it does not matter in which quadrant the angle $\theta$ lies or terminates. The value of product of all six trigonometric functions is 1. Therefore option (b) is correct.