exact value of cos 135
Understand the Problem
The question is asking for the exact value of the cosine function for the angle 135 degrees. This requires knowledge of trigonometric values for specific angles.
Answer
$-\frac{\sqrt{2}}{2}$
Answer for screen readers
The exact value of $\cos(135^\circ)$ is $-\frac{\sqrt{2}}{2}$.
Steps to Solve
- Identify the angle in the correct quadrant
The angle $135^\circ$ is in the second quadrant, where the cosine function is negative.
- Relate the angle to a known reference
The reference angle for $135^\circ$ can be calculated as follows: $$ 180^\circ - 135^\circ = 45^\circ $$ So, the reference angle is $45^\circ$.
- Use the known value of cosine for the reference angle
The cosine of the reference angle $45^\circ$ is: $$ \cos(45^\circ) = \frac{\sqrt{2}}{2} $$
- Apply the sign of cosine in the second quadrant
Since $135^\circ$ is in the second quadrant, where cosine is negative: $$ \cos(135^\circ) = -\cos(45^\circ) $$
- Final calculation
Thus, we can express the cosine of $135^\circ$ as: $$ \cos(135^\circ) = -\frac{\sqrt{2}}{2} $$
The exact value of $\cos(135^\circ)$ is $-\frac{\sqrt{2}}{2}$.
More Information
The angle $135^\circ$ is often used in trigonometry because it helps illustrate how angles in different quadrants affect the signs of trigonometric functions. The reference angle provides a simple way to determine the cosine value by relating it back to a known angle.
Tips
- Forgetting to consider the sign of the cosine function in the second quadrant.
- Confusing the reference angle with the original angle.
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