cos 150 exact value
Understand the Problem
The question is asking for the exact value of the cosine of 150 degrees. To find this value, we can use the unit circle or trigonometric identities.
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
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Identify the Reference Angle The angle of 150 degrees is in the second quadrant. To find the reference angle, we subtract 150 degrees from 180 degrees:
$$ 180^\circ - 150^\circ = 30^\circ $$ -
Determine the Cosine Sign in the Quadrant In the second quadrant, the cosine value is negative.
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Use the Cosine Value of the Reference Angle The cosine of the reference angle (30 degrees) is given by:
$$ \cos(30^\circ) = \frac{\sqrt{3}}{2} $$ -
Apply the Sign from the Quadrant Since the cosine is negative in the second quadrant, we have:
$$ \cos(150^\circ) = -\cos(30^\circ) = -\frac{\sqrt{3}}{2} $$
The exact value of the cosine of 150 degrees is $-\frac{\sqrt{3}}{2}$.
More Information
The cosine function describes the x-coordinate of a point on the unit circle. Since 150 degrees is an angle in the second quadrant, it has a negative cosine value despite the reference angle being a commonly known value. The angle 150 degrees is complementary to 30 degrees in terms of reference, making this calculation relatively straightforward.
Tips
- Misidentifying the quadrant and the sign of the cosine. It's important to remember that cosine is negative in the second quadrant.
- Confusing the reference angle calculation; always ensure to subtract the angle from 180 degrees for angles in the second quadrant.
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