Cos 150 Degrees Without Calculator

How to Calculate cos 150° Without a Calculator

The cosine of 150 degrees can be found using trigonometric identities and the unit circle. Since 150° is in the second quadrant, its cosine value will be negative. Here's how to compute it:

1. Identify the reference angle for 150°.
2. Use the cosine of the reference angle.
3. Apply the sign rule for the second quadrant.

The Formula for cos 150 Degrees

The cosine of 150 degrees can be calculated using the reference angle formula:

In the second quadrant, cosine values are negative, which is why we use a negative sign.

Step-by-Step Calculation

1. Find the reference angle: 180° - 150° = 30°.
2. Recall that cos(30°) = √3/2 ≈ 0.8660.
3. Apply the sign rule: cos(150°) = -cos(30°) = -√3/2 ≈ -0.8660.

Worked Example

Let's calculate cos(150°):

1. Reference angle: 180° - 150° = 30°.
2. cos(30°) = √3/2 ≈ 0.8660.
3. cos(150°) = -cos(30°) = -√3/2 ≈ -0.8660.

The exact value is -√3/2, and the approximate decimal value is -0.8660.

Interpreting the Result

The cosine of 150 degrees is -√3/2, which means:

• The x-coordinate on the unit circle is -√3/2.
• The angle 150° is in the second quadrant where cosine values are negative.
• The reference angle of 30° gives us the magnitude of the cosine value.