What is the exact value of cos 120?
Understand the Problem
The question is asking for the exact value of the cosine function at an angle of 120 degrees. This involves understanding the properties of trigonometric functions, particularly in relation to the unit circle.
Answer
The exact 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
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Convert the angle to radians
To use the cosine function, it can be helpful to convert the angle from degrees to radians.
$$ 120^\circ = \frac{120 \pi}{180} = \frac{2\pi}{3} \text{ radians} $$ -
Identify the quadrant
The angle $120^\circ$ is in the second quadrant, where the cosine function is negative. -
Determine the reference angle
The reference angle for $120^\circ$ can be found by subtracting it from $180^\circ$.
$$ 180^\circ - 120^\circ = 60^\circ $$ -
Use the cosine of the reference angle
Now, we can find the cosine value. The cosine of the reference angle $60^\circ$ is:
$$ \cos(60^\circ) = \frac{1}{2} $$ -
Apply the sign for the second quadrant
Since we are in the second quadrant, the cosine value will be negative:
$$ \cos(120^\circ) = -\cos(60^\circ) = -\frac{1}{2} $$
The exact value of $\cos(120^\circ)$ is $-\frac{1}{2}$.
More Information
The cosine value of $120^\circ$ reflects its position in the second quadrant. The angle corresponds to the reference angle of $60^\circ$, where the cosine of $60^\circ$ is known to be $\frac{1}{2}$. Therefore, the sign changes in the second quadrant lead to the negative result.
Tips
- Confusing degrees and radians: Make sure to convert degrees to radians if working in a context that requires it.
- Incorrect reference angle: Ensure you correctly identify the reference angle based on the quadrant.
