What is the exact value of cos(150°)?
Understand the Problem
The question requires us to find the exact value of cos(150°). This involves understanding trigonometric functions and special angles to determine the cosine of the given angle.
Answer
$-\frac{\sqrt{3}}{2}$
Answer for screen readers
$cos(150^\circ) = -\frac{\sqrt{3}}{2}$
Steps to Solve
- Find the reference angle
Since $150^\circ$ is in the second quadrant, the reference angle is found by subtracting it from $180^\circ$:
$180^\circ - 150^\circ = 30^\circ$
- Determine the cosine of the reference angle
The cosine of $30^\circ$ is a known value:
$cos(30^\circ) = \frac{\sqrt{3}}{2}$
- Determine the sign of the cosine in the second quadrant
In the second quadrant, cosine is negative.
- Apply the sign to the cosine of the reference angle
Therefore, $cos(150^\circ) = -cos(30^\circ)$
$cos(150^\circ) = -\frac{\sqrt{3}}{2}$
$cos(150^\circ) = -\frac{\sqrt{3}}{2}$
More Information
The angle $150^\circ$ is located in the second quadrant, and its reference angle is $30^\circ$. The cosine function is negative in the second quadrant. Therefore, $cos(150^\circ)$ is equal to the negative value of $cos(30^\circ)$, which is $-\frac{\sqrt{3}}{2}$.
Tips
A common mistake is forgetting that cosine is negative in the second quadrant and incorrectly stating the answer as $\frac{\sqrt{3}}{2}$. Another common mistake is using the wrong special triangle (e.g., using values for 45 degrees instead of 30 degrees).
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