cos 150 degrees in fraction
Understand the Problem
The question is asking for the value of cos(150 degrees) expressed in fractional form. To solve this, we can use the unit circle or trigonometric identities to find the cosine of the angle.
Answer
$-\frac{\sqrt{3}}{2}$
Answer for screen readers
The final answer is $-\frac{\sqrt{3}}{2}$.
Steps to Solve
- Determine the reference angle
The angle $150^\circ$ is in the second quadrant. The reference angle is calculated as: $$ 180^\circ - 150^\circ = 30^\circ $$
- Use the cosine value of the reference angle
In the second quadrant, the cosine value is negative. The cosine of the reference angle $30^\circ$ is: $$ \cos(30^\circ) = \frac{\sqrt{3}}{2} $$
- Apply the sign based on the quadrant
Since $150^\circ$ is in the second quadrant, we have: $$ \cos(150^\circ) = -\cos(30^\circ) $$
Substituting the known value: $$ \cos(150^\circ) = -\frac{\sqrt{3}}{2} $$
The final answer is $-\frac{\sqrt{3}}{2}$.
More Information
The cosine of $150^\circ$ illustrates the behavior of cosine in different quadrants. Cosine is positive in the first and fourth quadrants, while it is negative in the second and third quadrants. Knowing reference angles can help simplify finding trigonometric values.
Tips
Common mistakes include:
- Forgetting to consider the quadrant when determining the sign of the cosine value.
- Confusing the reference angle calculation.
To avoid these mistakes, always identify which quadrant the angle is in and remember the associated signs for cosine.