Q. 132 - Precalculus Enhanced with Graphing Utilities

Question

Use the figure to approximate the value of the six trigonometric functions at t to the nearest tenth. Then use a calculator to approximate each of the six trigonometric functions at t. 

a)$t=2$

b)$t=4$

Step-by-Step Solution

Answer

(a) all trigonometric function values are the same in the unit circle and as well as in the calculator at an angle of $t=2$

(b)all trigonometric function values are the same in the unit circle and as well as in the calculator at an angle of  $t=4$ 

Step 1. Given information.

The diagram of unit circle :

Part (a) Step 1. Plot a right triangle of the angle of t = 2 at origin in the unit circle.

The graph of unit circle:

In this graph we can see that:

$$\begin{gathered} b=-0.416 \\ p=0.909 \\ h=1 \end{gathered}$$

Step 3. Trigonometric functions by the unit circle.

Look at the right triangle.

$$\begin{gathered} \sin \left(t\right)=\frac{p}{h} \\ \sin \left(2\right)=\frac{0.909}{1} \\ =0.909 \end{gathered}$$

Similarly,

$$\begin{gathered} \cos \left(t\right)=\frac{b}{h} \\ \cos \left(2\right)=\frac{-0.416}{1} \\ =-0.416 \end{gathered}$$

$$\begin{gathered} \tan \left(t\right)=\frac{p}{b} \\ \tan \left(2\right)=\frac{0.909}{-0.416} \\ =-2.18 \end{gathered}$$

$$\begin{gathered} \cos ec\left(t\right)=\frac{1}{\sin \left(t\right)} \\ \cos ec\left(2\right)=\frac{1}{\sin \left(2\right)} \\ =\frac{1}{0.909} \\ =1.10 \end{gathered}$$

$$\begin{gathered} sec\left(2\right)=\frac{1}{\cos \left(2\right)} \\ =\frac{1}{-0.416} \\ =-2.40 \end{gathered}$$

$$\begin{gathered} cot\left(2\right)=\frac{1}{\tan \left(2\right)} \\ =\frac{1}{-2.18} \\ =-0.458 \end{gathered}$$

Step 4. trigonometric functions value by the calculator:

According to the calculator, All trigonometric functions value at $t=2$ are

$$\begin{gathered} \sin \left(2\right)=0.909 \\ \cos \left(2\right)=-0.416 \\ \tan \left(2\right)=-2.18 \\ \cos ec\left(2\right)=1.10 \\ sec\left(2\right)=-2.40 \\ cot\left(2\right)=-0.458 \end{gathered}$$

trigonometric functions value by unit circle and calculator are the same.

Step 6. Part (b). Plot a right triangle of the angle of t = 4 at origin in the unit circle:

The graph is:

Here in the right triangle 

$$\begin{gathered} b=-0.65 \\ p=-0.75 \\ h=1 \end{gathered}$$

Step 6. Trigonometric functions by the unit circle

$$\begin{gathered} \sin \left(4\right)=\frac{-0.75}{1} \\ =-0.75 \end{gathered}$$

$$\begin{gathered} \cos \left(4\right)=\frac{-0.65}{1} \\ =-0.65 \end{gathered}$$

$$\begin{gathered} \tan \left(4\right)=\frac{-0.75}{-0.65} \\ =1.15 \end{gathered}$$

$$\begin{gathered} \cos ec\left(4\right)=\frac{1}{-0.75} \\ =-1.33 \end{gathered}$$

$$\begin{gathered} sec\left(4\right)=\frac{1}{-0.65} \\ =-1.54 \end{gathered}$$

$$\begin{gathered} cot\left(4\right)=\frac{1}{1.15} \\ =0.87 \end{gathered}$$

Step 7. trigonometric functions value by the calculator

According to the calculator, All trigonometric functions value at $t=4$ are:

$$\begin{gathered} \sin \left(4\right)=-0.75 \\ \cos \left(4\right)=-0.65 \\ \tan \left(4\right)=1.15 \\ \cos ec\left(4\right)=-1.33 \\ sec\left(4\right)=-1.54 \\ cot\left(4\right)=0.87 \end{gathered}$$

trigonometric functions value by unit circle and calculator are the same.