Find the exact value. cos 135°

Find the exact value. cos 135°

The Correct Answer and Explanation is:

The exact value of cos 135° is: −22\boxed{-\dfrac{\sqrt{2}}{2}}−22​​​

Explanation

To find the exact value of cos 135°, we first recognize that this angle lies in the second quadrant of the unit circle. In the second quadrant, cosine values are negative because cosine corresponds to the x-coordinate of a point on the unit circle, and x-values are negative in that region.

Next, we use the concept of a reference angle. A reference angle is the acute angle formed between the terminal side of a given angle and the x-axis. For 135°, its reference angle is: 180°−135°=45°180° – 135° = 45°180°−135°=45°

This means that the cosine of 135° has the same magnitude as the cosine of 45°, but with a negative sign since it is in the second quadrant.

From trigonometric values of special angles, we know: cos⁡45°=22\cos 45° = \dfrac{\sqrt{2}}{2}cos45°=22​​

Since 135° is in the second quadrant, we apply the negative sign: cos⁡135°=−cos⁡45°=−22\cos 135° = -\cos 45° = -\dfrac{\sqrt{2}}{2}cos135°=−cos45°=−22​​

This value is exact because it comes from a well-known special angle (45°), which is part of the 30°-60°-90° and 45°-45°-90° right triangle relationships on the unit circle.

This value can also be understood geometrically. On the unit circle, a 135° angle corresponds to a point in the second quadrant that forms a 45° angle with the negative x-axis. The coordinates of this point are: (−22,22)\left(-\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2}\right)(−22​​,22​​)

Since cosine is the x-coordinate, this confirms that: cos⁡135°=−22\cos 135° = -\dfrac{\sqrt{2}}{2}cos135°=−22​​

In summary, cosine values depend on both the angle’s reference and the quadrant in which it lies. Understanding the unit circle and reference angles is key to finding exact trigonometric values without a calculator.