Chapter 7, Problem 1CRT

(a)

To determine

To find:The value of 3 $\sin t$ .

Answer to Problem 1CRT

The value of

  $\sin t$ is $\frac{\sqrt{5}}{3}$ .

Explanation of Solution

Given information:

The point $P\left(x,y\right)$ shown in figure has y-coordinate $\frac{\sqrt{5}}{3}$ .

  

Calculation:

According to the diagram the coordinates of the point $P$ are $\left(\cos t,\sin t\right)$ .

As given the y-coordinate of the point $P$ is $\frac{\sqrt{5}}{3}$ .

This gives $\sin t=\frac{\sqrt{5}}{3}$ .

Therefore, the value of $\sin t$ is $\frac{\sqrt{5}}{3}$ .

(b)

To determine

To find:The value of $\cos t$ .

Answer to Problem 1CRT

The value of $\cos t$ is $\frac{2}{3}$ .

Explanation of Solution

Given information:

The point $P\left(x,y\right)$ shown in figure has y-coordinate $\frac{\sqrt{5}}{3}$ .

  

Formula used:

Use the formula $\cos t=\sqrt{1-{\sin }^{2}t}$ .

Calculation:

As calculated in part(a) the value of $\sin t$ is $\frac{\sqrt{5}}{3}$ .

Substitute the value of $\sin t$ in the formula.

  $\begin{array}{c}\cos t=\sqrt{1-{\sin }^{2}t}\\ =\sqrt{1-{\left(\frac{\sqrt{5}}{3}\right)}^2}\\ =\sqrt{1-\frac{5}{9}}\\ =\sqrt{\frac{4}{9}}\\ =\frac{2}{3}\end{array}$

Therefore, the value of $\cos t$ is $\frac{2}{3}$ .

(c)

To determine

To find:The value of $\tan t$ .

Answer to Problem 1CRT

The value of $\tan t$ is $\frac{\sqrt{5}}{2}$ .

Explanation of Solution

Given information:

The point $P\left(x,y\right)$ shown in figure has y -coordinate $\frac{\sqrt{5}}{3}$ .

  

Formula used:

Use the formula $\tan t=\frac{\sin t}{\cos t}$ .

Calculation:

As calculated in part(a) the value of $\sin t$ is $\frac{\sqrt{5}}{3}$ and in part(b) the value of $\cos t$ is $\frac{2}{3}$ .

Now substitute $\frac{\sqrt{5}}{3}$ for $\sin t$ and $\frac{2}{3}$ for $\cos t$ in the formula.

  $\begin{array}{c}\tan t=\frac{\sin t}{\cos t}\\ =\frac{\frac{\sqrt{5}}{3}}{\frac{2}{3}}\\ =\frac{\sqrt{5}}{2}\end{array}$

Therefore, the value of $\tan t$ is $\frac{\sqrt{5}}{2}$ .

(d)

To determine

To find:The value of $\csc t$ .

Answer to Problem 1CRT

The value of $\csc t$ is $\frac{3\sqrt{5}}{5}$ .

Explanation of Solution

Given information:

The point $P\left(x,y\right)$ shown in figure has y-coordinate $\frac{\sqrt{5}}{3}$ .

  

Formula used:

Use the formula $\csc t=\frac{1}{\sin t}$ .

Calculation:

As calculated in part(a) the value of $\sin t$ is $\frac{\sqrt{5}}{3}$ .

Now substitute $\frac{\sqrt{5}}{3}$ for $\sin t$ in the formula.

  $\begin{array}{c}\csc t=\frac{1}{\sin t}\\ =\frac{1}{\frac{\sqrt{5}}{3}}\\ =\frac{3}{\sqrt{5}}\\ =\frac{3\sqrt{5}}{5}\end{array}$

Therefore, the value of $\csc t$ is $\frac{3\sqrt{5}}{5}$ .