Question

Find an approximate value of he given trigonometric functions by using (a) the figure and (b) a calculator. Compare the two values.

$\cos 0.8$

Short answer

a. From the figure, approximate value of $\cos 0.8\approx 0.7$.

b. Both values are very close to each other.

Step-by-step solution

Part a. Step 1. State the concept.

For the given point on the unit circle$\cos t=x$ and$\sin t=y$ where$\left(x,y\right)$ is the terminal point to real number $t$.

Part a. Step 2. Use the figure to find the terminal point determined by the real number t. 

Draw the coordinates corresponding to $t=0.8$.

Since given point on circle lies in the first quadrant, therefore both $x$-axis and $y$-axis are positive and can be approximated to $\left(0.7,0.7\right)$.

Part a. Step 3. Find the value.

Since $\cos t=x$, therefore,$\cos 0.8=0.7$ because from the figure it can be observed that the $x$-coordinate is approximately $0.7$.

Part b. Step 1. State the concept.

For the given point on the unit circle$\cos t=x$ and$\sin t=y$ where$\left(x,y\right)$ is the terminal point to real number $t$.

Part b. Step 2. Find the value using calculator.

Set the calculator to radians first. Now calculate the value of $\cos 0.8$.

The output given by the calculator is $\cos 0.8=0.69670670934$.

Part b. Step 3. Compare the values.

Observe that from the graph, the approximated value of$\cos 0.8=0.7$ and from calculator value$\cos 0.8=0.69670670934$ which can be approximated to the same value. Therefore, both values are very close to each other.