Use the reference angle to find the exact value of the following expression. Show all your work step by step.
cos 135°
To find the exact value of cos 135° using the reference angle, we need to find the cosine of the reference angle and determine its sign.
Step 1: Find the reference angle.
The reference angle is the acute angle formed between the terminal side of the given angle (135°) and the x-axis.
Since 135° is in the second quadrant, the reference angle can be found by subtracting 135° from 180°:
Reference angle = 180° - 135° = 45°
Step 2: Determine the sign.
In the second quadrant, the cosine function is negative.
Since the reference angle is positive, the cosine of the reference angle will also be negative.
Step 3: Find the exact value of cos 135°.
Since the reference angle is 45° and the cosine function is negative in the second quadrant:
cos 135° = -cos 45°
Step 4: Find the exact value of cos 45°.
The cosine of 45° is equal to √2/2.
cos 45° = √2/2
Step 5: Substitute the value into the expression.
cos 135° = -cos 45° = -√2/2
Therefore, the exact value of cos 135° using the reference angle is -√2/2.