What is the exact value of cos 120 degrees?
Understand the Problem
The question is asking for the exact value of the cosine of 120 degrees, which is a trigonometric function. To solve this, we can use the unit circle or the cosine function in a triangle.
Answer
The value of $\cos(120^\circ)$ is $-\frac{1}{2}$.
Answer for screen readers
The exact value of $\cos(120^\circ)$ is $-\frac{1}{2}$.
Steps to Solve
- Identify the Angle in the Unit Circle
The angle of 120 degrees is in the second quadrant of the unit circle. The reference angle is calculated as follows:
$$ 180^\circ - 120^\circ = 60^\circ $$
- Determine the Cosine Value for the Reference Angle
The cosine of a reference angle gives the cosine value in the second quadrant. The cosine function value for the reference angle of $60^\circ$ is:
$$ \cos(60^\circ) = \frac{1}{2} $$
- Adjust for the Quadrant
In the second quadrant, cosine values are negative. Therefore, we need to apply this sign to the reference angle's cosine value:
$$ \cos(120^\circ) = -\cos(60^\circ) = -\frac{1}{2} $$
The exact value of $\cos(120^\circ)$ is $-\frac{1}{2}$.
More Information
The cosine function relates to the x-coordinate of a point on the unit circle. At 120 degrees, we are positioned in the second quadrant, where cosine values are negative. It’s important to remember that for angles in the second quadrant, we obtain negative values for cosine.
Tips
- A common mistake is forgetting that cosine is negative in the second quadrant. Always check which quadrant the angle is in and adjust the sign of the cosine value accordingly.
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