Finding the Values of tan 135 and cot 135
To calculate the values of tan 135 and cot 135, we can use the properties of trigonometric functions, specifically focusing on the unit circle and the reference angles.
Step 1: Understanding the Angle
The angle of 135 degrees is located in the second quadrant of the unit circle. In the second quadrant, the sine function is positive, and the cosine function is negative.
Step 2: Finding the Reference Angle
The reference angle for 135 degrees is calculated as follows:
- Reference angle = 180° – 135° = 45°
Step 3: Using Known Trigonometric Values
We know the values of the sine and cosine functions for the reference angle (45 degrees):
- sin(45°) = √2/2
- cos(45°) = √2/2
However, since we are in the second quadrant for 135 degrees:
- sin(135°) = sin(45°) = √2/2
- cos(135°) = -cos(45°) = -√2/2
Step 4: Calculating tan 135
The tangent of an angle is defined as the ratio of the sine to the cosine:
tan(135°) = sin(135°) / cos(135°)
Substituting the values, we get:
tan(135°) = (√2/2) / (-√2/2) = -1
Step 5: Calculating cot 135
The cotangent is the reciprocal of the tangent:
cot(135°) = 1 / tan(135°)
So:
cot(135°) = 1 / (-1) = -1
Final Values
In conclusion, we’ve calculated:
- tan 135 = -1
- cot 135 = -1