Find the exact value of sec 120.
Understand the Problem
The question is asking us to determine the exact value of the secant function at an angle of 120 degrees. To solve this, we can use the relationship between the secant function and the cosine, since secant is the reciprocal of cosine.
Answer
$-2$
Answer for screen readers
The exact value of the secant function at an angle of 120 degrees is $-2$.
Steps to Solve
- Identify the relationship between secant and cosine
To find the secant of an angle, we use the relationship:
$$ \sec(\theta) = \frac{1}{\cos(\theta)} $$
For this problem, we have $\theta = 120^\circ$.
- Calculate the cosine of 120 degrees
Next, we need to find $\cos(120^\circ)$. Since $120^\circ$ is in the second quadrant, the cosine is negative. The reference angle is
$$ 180^\circ - 120^\circ = 60^\circ $$
Thus,
$$ \cos(120^\circ) = -\cos(60^\circ) $$
And we know from the unit circle that:
$$ \cos(60^\circ) = \frac{1}{2} $$
So,
$$ \cos(120^\circ) = -\frac{1}{2} $$
- Calculate the secant of 120 degrees
Now, we can find the secant:
$$ \sec(120^\circ) = \frac{1}{\cos(120^\circ)} = \frac{1}{-\frac{1}{2}} $$
Therefore,
$$ \sec(120^\circ) = -2 $$
The exact value of the secant function at an angle of 120 degrees is $-2$.
More Information
The secant function provides information about the ratio of the hypotenuse to the adjacent side in a right triangle. It's also worth noting that secant is undefined at angles where cosine is zero, such as 90 degrees and 270 degrees.
Tips
- Confusing the angle's quadrant and getting the sign of cosine wrong. Remember that cosine is positive in the first and fourth quadrants and negative in the second and third.
- Miscalculating the reference angle. Be sure to accurately determine the reference angle based on the position of the original angle.