To find the values of tan 135° and cot 135°, we can use the unit circle and the properties of trigonometric functions.
Step 1: Understanding the angle
The angle 135° is in the second quadrant of the unit circle. In this quadrant, the sine function is positive while the cosine function is negative.
Step 2: Reference Angle
The reference angle for 135° is found by subtracting it from 180°:
180° – 135° = 45°
This means that the values of tan 135° and cot 135° can be derived from the corresponding values for the angle 45°.
Step 3: Finding tan 135°
Since we know that:
tan(θ) = sin(θ) / cos(θ)
For 135°, we have:
tan 135° = tan(180° – 45°) = -tan 45°
We know that:
tan 45° = 1
Therefore:
tan 135° = -1
Step 4: Finding cot 135°
The cotangent function is the reciprocal of the tangent function:
cot(θ) = 1 / tan(θ)
So for cot 135°:
cot 135° = 1 / tan 135° = 1 / (-1)
Thus:
cot 135° = -1
Conclusion
In summary, we find that:
- tan 135° = -1
- cot 135° = -1