Solve the equation 2sin(x) + 1 = 0.
1 answer:
Answer:
- 3rd quadrant: 210°
- 4th quadrant: 330°
Step-by-step explanation:
The equation can be rearranged by subtracting 1, then dividing by 2. Doing so gives you ...
... sin(x) = -1/2
The sine of an angle is the y-coordinate of the point where its terminal ray intersects the unit circle. A horizontal line at y=-1/2 intersects the unit circle in two places. (Refer to the attached diagram.) In each case, the reference angle (the smallest angle made with the x-axis) is 30°.
Conventionally, we measure angles counterclockwise from the +x axis, so the 3rd-quadrant angle (between 180° and 270°) will be 180°+30° = 210°.
The 4th-quadrant angle (between 270° and 360°) will be 360°-30° = 330°.
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<em>Comment on the diagram</em>
The geometry program used to create the figure decided to show the angles measured clockwise. To get the answers you want, you need to subtract the angles shown from 360°.
<em>Comment on calculator solutions</em>
If you use a calculator (in degrees mode) to find the angle whose sine is -1/2, it will tell you sin⁻¹(-0.5) is -30°. This means the angle is 30° measured clockwise from the +x axis. Of course, the value you want in that quadrant is 360°-30° = 330°. You have to understand that the third quadrant angle can be found by adding the reference angle (30°) to 180°.
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