Chapter 7, Problem 4CRT

To determine

Tofind:The values of $\sin t$ , $\tan t$ , $\cot t$ , $\sec t$ and $\csc t$ .

Answer to Problem 4CRT

The values of $\sin t$ , $\tan t$ , $\cot t$ , $\sec t$ and $\csc t$ are $-\frac{24}{25}$ , $-\frac{24}{7}$ , $-\frac{7}{24}$ , $\frac{25}{7}$ and $-\frac{25}{24}$ respectively.

Explanation of Solution

Given information:

The value $\cos t=\frac{7}{25}$ and $\tan t<0$ .

Calculation:

As given $\cos t=\frac{7}{25}$ , then the value of adjacent is $7$ and hypotenuse is $25$ .

Use the Pythagoras theorem.

  $\begin{array}{c}\text{opp}=\sqrt{{\left(\text{hyp}\right)}^2-{\left(\text{adj}\right)}^2}\\ =\sqrt{{25}^2-7^2}\\ =\sqrt{625-49}\\ =\sqrt{576}\\ =24\end{array}$

As the value of adjacent, opposite and hypotenuse is $7$ , $24$ and $25$ respectively.

Use the formula for tangent.

  $\begin{array}{c}\tan t=\frac{\text{opp}}{\text{adj}}<0\\ =-\frac{24}{7}\end{array}$

Use the formula for sine.

  $\begin{array}{c}\sin t=\tan t.\cos t\\ =\left(-\frac{24}{7}\right)\times \frac{7}{25}\\ =-\frac{24}{25}\end{array}$

Use the formula for cotangent.

  $\begin{array}{c}\cot t=\frac{1}{\tan t}\\ =\frac{1}{-\frac{24}{7}}\\ =-\frac{7}{24}\end{array}$

Use the formula for secant.

  $\begin{array}{c}\sec t=\frac{1}{\cos t}\\ =\frac{1}{\frac{7}{25}}\\ =\frac{25}{7}\end{array}$

Use the formula for cosecant.

  $\begin{array}{c}\csc t=\frac{1}{\sin t}\\ =\frac{1}{-\frac{24}{25}}\\ =-\frac{25}{24}\end{array}$

Therefore, the values of $\sin t$ , $\tan t$ , $\cot t$ , $\sec t$ and $\csc t$ are $-\frac{24}{25}$ , $-\frac{24}{7}$ , $-\frac{7}{24}$ , $\frac{25}{7}$ and $-\frac{25}{24}$ respectively.