How to Work Out Sin 150 Without Calculator

Calculating sin 150 degrees without a calculator requires understanding of trigonometric identities and reference angles. This guide explains the methods and provides a step-by-step calculation.

Understanding sin 150 Degrees

The sine function is periodic with a period of 360 degrees, meaning sin(θ) = sin(θ + 360° × n) for any integer n. This property allows us to find equivalent angles within the first period (0° to 360°).

sin 150° is equivalent to sin(180° + 30°), which places it in the third quadrant where sine values are negative.

To find sin 150°, we can use reference angles and trigonometric identities. The reference angle for 150° is 30° (180° - 150°).

Using Reference Angle

The reference angle is the acute angle that the terminal side of a given angle makes with the x-axis. For 150°:

Reference angle = 180° - 150° = 30°

Since 150° is in the second quadrant where sine is positive, we can use the reference angle to find sin 150°.

Using Trigonometric Identities

We can use the sine of a sum identity to express sin 150°:

sin(150°) = sin(180° - 30°) = sin 180° cos 30° - cos 180° sin 30°

We know that sin 180° = 0 and cos 180° = -1, so the equation simplifies to:

sin(150°) = 0 × cos 30° - (-1) × sin 30° = sin 30°

Therefore, sin 150° = sin 30° = 0.5.

Step-by-Step Calculation

Identify the quadrant: 150° is in the second quadrant (90° to 180°).
Find the reference angle: 180° - 150° = 30°.
Recall that sine is positive in the second quadrant.
Use the sine of a sum identity: sin(150°) = sin(180° - 30°).
Apply the identity: sin(180° - 30°) = sin 180° cos 30° - cos 180° sin 30°.
Substitute known values: sin 180° = 0, cos 180° = -1, sin 30° = 0.5, cos 30° = √3/2.
Simplify: sin(150°) = 0 × √3/2 - (-1) × 0.5 = 0.5.

Note: The result is positive because sine is positive in the second quadrant.

Verification

To verify our calculation, we can use the unit circle or a calculator:

sin(150°) ≈ 0.5

This matches our manual calculation, confirming the result.

Frequently Asked Questions

Why is sin 150° positive?

sin 150° is positive because 150° is in the second quadrant where sine values are positive. The reference angle is 30°, and sin 30° is 0.5.

Can I use a calculator to verify sin 150°?

Yes, using a calculator, you can input 150 degrees in the sine function to verify that the result is approximately 0.5.

What is the reference angle for 150°?

The reference angle for 150° is 30° (180° - 150°). This is the acute angle that helps determine the sine value.

How do I calculate sin 150° using identities?

You can use the identity sin(180° - θ) = sin θ. For θ = 30°, sin(150°) = sin(180° - 30°) = sin 30° = 0.5.