Question

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

$\tan (-1.3)$

Short answer

a. From the figure, the approximate value of $\mathbf{tan}\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{.3}\mathbf{)}\mathbf{\approx }\mathbf{-}\mathbf{3}\mathbf{.6}$.

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 $t=-1.3$.

Since the given point on the circle lies in the fourth quadrant, therefore, $x$-coordinate is positive and $y$-coordinate is negative and can be approximated to$\left(0.27,-0.97\right)$.

Part a. Step 3. Find the value.

Since $\tan t=\frac{y}{x}$, therefore, $\tan \left(-1.3\right)\approx -3.6$ because from the figure it can be observed that the $x$-coordinate is 0.27 and $y$-coordinate both are approximately $-0.97$.

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 $\tan \left(-1.3\right)$.

The output given by the calculator is $\tan \left(-1.3\right)=-3.60210244797$.

Part b. Step 3. Compare the values.

Observe that from the graph, the approximated value of $\tan \left(-1.3\right)\approx -3.6$ and from calculator value $\tan \left(-1.3\right)=-3.60210244797$ which can be approximated to the same value. Therefore, both values are very close to each other.