Question

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

$\sin 1.2$

Short answer

a. From the figure, the approximate value of $\mathbf{sin}\mathbf{1}\mathbf{\approx }\mathbf{0}\mathbf{.9}$.

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 a 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 $t=1.2$.

Since the given point on the circle lies in the first quadrant, therefore both $x$-axis and $y$-axis are positive and can be approximated to$\left(0.4,0.9\right)$.

Part a. Step 3. Find the value.

Since $\sin t=y$, therefore, $\sin 1.2=0.9$ because from the figure, it can be observed that the $y$-coordinate is approximately 0.9.

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 a real number $t$.

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

Set the calculator to radians first. Now calculate the value of $\sin 1.2$.

The output given by the calculator is $\sin 1.2=0.93203908596$.

Part b. Step 3. Compare the values.

Observe that from the graph, the approximated value of $\sin 1.2=0.9$ and from calculator value $\sin 1.2=0.93203908596$ which can be approximated to the same value. Therefore, both values are very close to each other.