Q. 131 - Precalculus Enhanced with Graphing Utilities

Question

In Problem$131$, 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) $t=1$

(b) $t=5.1$

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 $t=1$

$$\begin{gathered} \sin 1=0.8 \\ \cos 1=0.5 \\ \tan 1=1.6 \\ csc 1=1.6 \\ sec 1=2 \\ cot 1=0.6 \end{gathered}$$

(b)all trigonometric function values are the same in the unit circle and as well as in the calculator at an angle of  $t=5.1$

$$\begin{gathered} \sin (5.1)=-0.9 \\ \cos (5.1)=0.4 \\ \tan (5.1)=-2.3 \\ csc(5.1)=-1.1 \\ sec(5.1)=2.5 \\ cot(5.1)=-0.4 \end{gathered}$$

Step 1. Given information

Values of angle

$$\begin{gathered} t=1 \\ t=5.1 \end{gathered}$$

unit circle 

Step 2. Part (a)

plot a right triangle of the angle of $t=1$ at origin in the unit circle

The graph state that the coordinate of Point A is$(0.54,0.84)$

so$$\begin{gathered} a=0.54 \\ b=0.84 \\ c=1 \end{gathered}$$

Step 3. Trigonometric functions by the unit circle

Sine function using the triangle in the unit circle

$$\begin{gathered} \sin \left(t\right)=\frac{opposite}{hypotenues} \\ \sin \left(1\right)=\frac{0.8}{1} \\ \sin \left(1\right)=0.8 \end{gathered}$$

Similarly

$$\begin{gathered} \cos \left(t\right)=\frac{hypotenues}{opposite} \\ \cos \left(1\right)=\frac{0.5}{1} \\ =0.5 \end{gathered}$$

Step 4. Trigonometric functions by the unit circle

Using trigonometric identities

$$\begin{gathered} \tan \left(t\right)=\frac{\sin \left(t\right)}{\cos \left(t\right)} \\ \tan \left(1\right)=\frac{0.8}{0.5} \\ =1.6 \\ csc \left(1\right)=\frac{1}{\sin \left(1\right)} \\ =\frac{1}{0.8} \\ =1.3 \\ sec \left(1\right)=\frac{1}{\cos \left(1\right)} \\ =\frac{1}{0.5} \\ =2 \\ cot \left(1\right)=\frac{1}{\tan 1} \\ =\frac{1}{1.6} \\ =0.6 \end{gathered}$$

Step 5. trigonometric functions value by the calculator

According to the calculator, All trigonometric functions value at $t=1$ are

$$\begin{gathered} \sin 1=0.8 \\ \cos 1=0.5 \\ \tan 1=1.6 \\ csc 1=1.6 \\ sec 1=2 \\ cot 1=0.6 \end{gathered}$$

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

Step 6. Part (b)

plot a right triangle of the angle of $t=5.1$ at origin in the unit circle

The graph state that the coordinate of Point A is $(0.4,-0.9)$

so

$$\begin{gathered} a=0.4 \\ b=-0.9 \\ c=1 \end{gathered}$$

Step 7. Trigonometric functions by the unit circle

Sine function using the triangle in the unit circle

$$\begin{gathered} \sin \left(t\right)=\frac{opposite}{hypotenues} \\ \sin (5.1)=\frac{-0.9}{1} \\ \sin \left(1\right)=-0.9 \end{gathered}$$ 

Similarly 

$$\begin{gathered} \cos \left(t\right)=\frac{hypotenues}{opposite} \\ \cos (5.1)=\frac{0.4}{1} \\ =0.4 \end{gathered}$$

Step 8. Trigonometric functions by the unit circle

Using trigonometric identities 

$$\begin{gathered} \tan \left(t\right)=\frac{\sin \left(t\right)}{\cos \left(t\right)} \\ \tan (5.1)=\frac{-0.9}{0.4} \\ =-2.25 \\ csc (5.1)=\frac{1}{-0.9} \\ =\frac{1}{-0.9} \\ =-1.1 \\ sec (5.1)=\frac{1}{0.4} \\ =\frac{1}{0.5} \\ =2.5 \\ cot (5.1)=\frac{1}{\tan 1} \\ =\frac{1}{1.6} \\ =-0.4 \end{gathered}$$

Step 9. trigonometric functions value by the calculator

According to the calculator, All trigonometric functions value at $t=5.1$ are

$$\begin{gathered} \sin (5.1)=-0.9 \\ \cos (5.1)=0.4 \\ \tan (5.1)=-2.3 \\ csc(5.1)=-1.1 \\ sec(5.1)=2.5 \\ cot(5.1)=-0.4 \end{gathered}$$

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