exact value of cos 150
Understand the Problem
The question is asking for the exact value of the cosine of 150 degrees, which involves understanding trigonometric values for specific angles.
Answer
$-\frac{\sqrt{3}}{2}$
Answer for screen readers
The exact value of the cosine of 150 degrees is $-\frac{\sqrt{3}}{2}$.
Steps to Solve
- Identify the Reference Angle
The angle of 150 degrees is in the second quadrant. To find the reference angle, subtract 150 degrees from 180 degrees.
$$ 180^\circ - 150^\circ = 30^\circ $$
- Determine the Cosine Value
The cosine of an angle in the second quadrant is negative. The cosine of the reference angle (30 degrees) is known.
$$ \cos(30^\circ) = \frac{\sqrt{3}}{2} $$
- Apply the Sign based on Quadrant
Since 150 degrees is in the second quadrant, the cosine will be negative:
$$ \cos(150^\circ) = -\cos(30^\circ) = -\frac{\sqrt{3}}{2} $$
The exact value of the cosine of 150 degrees is $-\frac{\sqrt{3}}{2}$.
More Information
The cosine function reflects the horizontal component of a point on the unit circle. For 150 degrees, the angle corresponds to a point on the unit circle that lies in the second quadrant where the x-coordinates (cosine values) are negative. The reference angle of 30 degrees helps us find its value easily.
Tips
- Positive and Negative Quadrants: Students often forget that cosine is negative in the second quadrant. Always double-check the quadrant to determine the sign of the cosine value.
- Reference Angle Confusion: Ensure you find the correct reference angle, as mistakes here can lead to incorrect final values.
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