Q. 87 - Precalculus Enhanced with Graphing Utilities

Question

Find the exact value of: $\sin 40°+\sin 130°+\sin 220°+\sin 310°$

Step-by-Step Solution

Answer

The value of $\sin 40°+\sin 130°+\sin 220°+\sin 310°=0$

Step 1. Given information

The given value is $\sin 40°+\sin 130°+\sin 220°+\sin 310°$

Step 2. Sketch the graph of sin 130 ° for reference angle α

The reference angle for $\sin 130° is 50°$

Step 3. Sketch the graph of sin 220 ° for reference angle

The reference angle for $\sin 220°is 40°$

Step 4. Sketch the graph of sin 310 ° for reference angle

The reference angle for $\sin 310° is 50°$

Step 5. Finding the value of sin 40 ° + sin 130 ° + sin 220 ° + sin 310 °

If $\alpha$ is any reference angle for $\theta$ then,

$\sin \theta =\pm \sin \alpha$

so,$\sin 130°=\pm \sin 50°,\sin 220°=\pm \sin 40°,and \sin 310°=\pm \sin 50°$

Now check in which quadrant the given angles lie

$\sin 40°$  lies in quadrant-I,where is positive

$\sin 130°$  lies in quadrant-II,where is positive

 $\sin 220°$ lies in quadrant-III,where is negative

$\sin 310°$  lies in quadrant-IV,where is negative

Replace the given angle in the expression with the respective reference angle

$$\begin{gathered} \sin 40°+\sin 130°+\sin 220°+\sin 310°=\sin 40°+\sin 50°+(-\sin 40°)+(-\sin 50°) \\ =(\sin 40°-\sin 40°)+(\sin 50°-\sin 50°) \\ =0 \end{gathered}$$

Therefore,the value of $\sin 40°+\sin 130°+\sin 220°+\sin 310°=0$