Evaluate Cos 135 Degrees Without Using A Calculator

Understanding Cosine

The cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse. On the unit circle, cosine represents the x-coordinate of a point at a given angle from the positive x-axis.

For angles beyond 90°, we use reference angles and the unit circle to determine cosine values.

Reference Angle Method

The reference angle is the acute angle that the terminal side of a given angle makes with the x-axis. For 135°, which is in the second quadrant:

1. Identify the quadrant: 135° is between 90° and 180° (second quadrant).
2. Calculate the reference angle: 180° - 135° = 45°.
3. Determine the sign: In the second quadrant, cosine is negative.
4. Use the reference angle's cosine value: cos 45° = √2/2 ≈ 0.7071.
5. Apply the sign: cos 135° = -cos 45° = -√2/2 ≈ -0.7071.

Unit Circle Approach

The unit circle has a radius of 1 and is centered at the origin (0,0). Any angle θ corresponds to a point (cos θ, sin θ) on the unit circle.

For 135°:

1. Locate 135° on the unit circle (second quadrant).
2. Identify the coordinates: The x-coordinate is cos 135°, and the y-coordinate is sin 135°.
3. Since 135° is 45° from the negative x-axis, the coordinates are (-cos 45°, sin 45°).
4. Thus, cos 135° = -cos 45° = -√2/2 ≈ -0.7071.

Practical Example

Let's verify cos 135° using both methods:

Reference Angle Method

1. Reference angle = 180° - 135° = 45°.
2. cos 45° = √2/2 ≈ 0.7071.
3. cos 135° = -cos 45° ≈ -0.7071.

Unit Circle Method

1. 135° is in the second quadrant.
2. Coordinates: (-cos 45°, sin 45°).
3. cos 135° = -cos 45° ≈ -0.7071.

Both methods yield the same result, confirming cos 135° = -√2/2 ≈ -0.7071.

Common Mistakes

When evaluating cosine without a calculator, common errors include:

• Forgetting to apply the negative sign in the second quadrant.
• Using the wrong reference angle (e.g., 135° - 90° instead of 180° - 135°).
• Confusing cosine with sine or tangent values.
• Assuming cosine is always positive.