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. Itâ€™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.