Chapter 4, Problem 25CR

To determine

To find:thesix trigonometric functions at the real number $t=4\pi$ if it is possible.

Answer to Problem 25CR

The six trigonometric functions are $\sin 4\pi =0$ , $\cos 4\pi =1$ , $\tan 4\pi =0$ , $\csc 4\pi =\text{ Undefined}$ , $\sec 4\pi =1$ , and $\cot 4\pi =\text{ Undefined}$ .

Explanation of Solution

Definition used:

Let t be a real number and let $\left(x,y\right)$ be the point on the unit circle corresponding to t.

  $\begin{array}{l}\sin t=y\cos t=x\tan t=\frac{y}{x},x\ne 0\\ \csc t=\frac{1}{y},y\ne 0\sec t=\frac{1}{x}\text{,}x\ne 0\cot t=\frac{x}{y},y\ne 0\end{array}$

Calculation:

For every real number t, there exist the corresponding to the point $\left(x,y\right)$ on the unit circle.

Here, two angles $t=4\pi$ and $t=2\pi$ are coterminal.

From Figure 4.13 on page 268, it is observed that the real number $t=2\pi$ corresponding to the point $\left(1,0\right)$ .

That is, the real number $t=4\pi$ corresponding to the point $\left(1,0\right)$ on the unit circle.

Thus, the values of $x=1\text{ and }y=0$

The six trigonometric function at the real number is computed as follows,

Substitute $t=4\pi$ and $y=0$ in $\sin t=y$ , the trigonometric function $\sin 4\pi =0$ .

Substitute $t=4\pi$ and $x=1$ in $\cos t=x$ , the trigonometric function $\cos 4\pi =1$ .

Substitute $t=4\pi$ , $x=1$ , and $y=0$ in $\tan t=\frac{y}{x}$ , the trigonometric function is,

  $\begin{array}{c}\tan 4\pi =\frac{0}{1}\\ =0\end{array}$

That is, the trigonometric function $\tan 4\pi =0$ .

The trigonometric function $\csc t=\frac{1}{y}$ defined for $y\ne 0$ but the value of y zero.

Therefore, the trigonometric function $\csc 4\pi$ is undefined.

Substitute $t=4\pi$ and $x=1$ in $\sec t=\frac{1}{x}$ , the trigonometric function is,

  $\begin{array}{c}\sec 4\pi =\frac{1}{1}\\ =1\end{array}$

That is, the trigonometric function $\sec 4\pi =1$ .

The trigonometric function $\cot t=\frac{x}{y}$ defined for $y\ne 0$ but the value of y zero.

Therefore, the trigonometric function $\cot 4\pi$ is undefined.

Therefore, the six trigonometric functions are $\sin 4\pi =0$ , $\cos 4\pi =1$ , $\tan 4\pi =0$ , $\csc 4\pi =\text{ Undefined}$ , $\sec 4\pi =1$ , and $\cot 4\pi =\text{ Undefined}$ .