To find the exact value of
cos 132 degrees, we can follow a few steps using the unit circle and trigonometric identities.
First, we can recognize that the angle 132 degrees is located in the second quadrant of the unit circle, where the cosine function takes negative values.
We can express 132 degrees in relation to a reference angle. The reference angle for 132 degrees is:
Reference angle = 180 degrees – 132 degrees = 48 degrees.
Now, we can use the cosine of the reference angle to find the cosine of 132 degrees:
cos 132 degrees = -cos 48 degrees
Next, to find the exact value of cos 48 degrees, we can use its approximation, but it’s important to note that cos 48 doesn’t have a simple exact value like some standard angles. Instead, we can express it in terms of sine or use an approximate value. However, the exact value of cos 48 degrees can be found using either trigonometric tables or a calculator, and it is approximately:
cos 48 degrees ≈ 0.6691.
Now substituting back to find cos 132 degrees:
cos 132 degrees ≈ -0.6691.
In conclusion, the exact value of cos 132 degrees is:
cos 132 degrees ≈ -0.6691
This computation not only shows how to derive the value using trigonometric identities but also emphasizes understanding the properties of angles within different quadrants.