Chapter 4.4, Problem 32E

To determine

The exact values of the six trigonometric functions of $\theta$ with the given constraint.

Answer to Problem 32E

The exact values of the six trigonometric functions of $\theta$ are

  $\begin{array}{l}\text{sin}\theta \text{=}-1\\ \text{cos}\theta \text{=}0\\ \text{tan}\theta \text{=not defined}\\ \text{sec}\theta \text{=not defined}\\ \text{csc}\theta \text{=}-1\\ \text{cot}\theta =\text{0}\end{array}$

Explanation of Solution

Given info:

  $\text{tan}\theta \text{ is undefined and }\pi \le \theta \le 2\pi$

Formula used:

In quadrant I all trigonometric functions are positive.

In quadrant IIsine functions are positive.

In quadrant IIItangent functions are positive.

In quadrant IVcosine functions are positive.

  

  $\begin{array}{l}r=\sqrt{x^2+y^2}\\ \text{sin}\theta \text{=}\frac{y}{r}\\ \text{cos}\theta \text{=}\frac{x}{r}\\ \text{tan}\theta \text{=}\frac{y}{x}\\ \text{csc}\theta \text{=}\frac{r}{y}\\ \text{sec}\theta \text{=}\frac{r}{x}\\ \text{cot}\theta \text{=}\frac{x}{y}\end{array}$

Calculation:

Given that $\text{tan}\theta \text{ is undefined}$

Hence angle $\theta ={90}^0$ or $\theta ={270}^0$

And $\pi \le \theta \le 2\pi$

The common angle is $\theta ={270}^0$ .

So $\theta ={270}^0$ .

Now we have

  $\begin{array}{l}\text{sin}\theta \text{=sin270=}-1\\ \text{cos}\theta \text{=cos270=}0\\ \text{tan}\theta \text{=}\tan \text{270=not defined}\\ \text{sec}\theta \text{=}\sec \text{270=not defined}\\ \text{csc}\theta \text{=}\csc \text{270=}-1\\ \text{cot}\theta \text{=}\cot \text{270}=\text{0}\end{array}$

Conclusion:

Thus,the exact values of the six trigonometric functions of $\theta$ are

  $\begin{array}{l}\text{sin}\theta \text{=}-1\\ \text{cos}\theta \text{=}0\\ \text{tan}\theta \text{=not defined}\\ \text{sec}\theta \text{=not defined}\\ \text{csc}\theta \text{=}-1\\ \text{cot}\theta =\text{0}\end{array}$ .