What is the exact value of cos(150°)?
Understand the Problem
The question asks to find the exact value of cos(150°). This means we need to determine the cosine of 150 degrees without using a calculator, likely by using trigonometric identities or reference angles.
Answer
$-\frac{\sqrt{3}}{2}$
Answer for screen readers
$-\frac{\sqrt{3}}{2}$
Steps to Solve
- Find the reference angle
The reference angle is the acute angle formed by the terminal side of the given angle and the x-axis. Since $150^\circ$ is in the second quadrant, we find the reference angle by subtracting it from $180^\circ$:
$180^\circ - 150^\circ = 30^\circ$
- Determine the cosine of the reference angle
We know that $\cos(30^\circ) = \frac{\sqrt{3}}{2}$.
- Determine the sign of the cosine in the second quadrant
In the second quadrant, the x-coordinate is negative, and cosine corresponds to the x-coordinate on the unit circle. Therefore, cosine is negative in the second quadrant.
- Apply the sign to the cosine of the reference angle
Since cosine is negative in the second quadrant, we have $\cos(150^\circ) = -\cos(30^\circ)$. Therefore,
$\cos(150^\circ) = -\frac{\sqrt{3}}{2}$
$-\frac{\sqrt{3}}{2}$
More Information
The angle $150^\circ$ is a special angle because its reference angle, $30^\circ$, is a special angle. The special angles $30^\circ$, $45^\circ$, and $60^\circ$ have well-known trigonometric ratios that are easy to remember and frequently used.
Tips
A common mistake is forgetting to check the quadrant of the angle and thus getting the sign wrong. A quick sketch of the angle on the coordinate plane can help avoid this mistake. Another mistake is confusing the values of $\sin(30^\circ)$ and $\cos(30^\circ)$.
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