How to Calculate Sin 165 Degrees

Introduction

The sine of an angle in the unit circle represents the y-coordinate of the corresponding point. For 165 degrees, which is in the second quadrant, we can use trigonometric identities to find its sine value.

Key points to remember:

• 165° is in the second quadrant where sine values are positive
• The reference angle for 165° is 15° (180° - 165°)
• We can use the identity sin(180° - θ) = sinθ

Method: Using Reference Angles

The most straightforward method to calculate sin 165° is by using reference angles. Here's how it works:

1. Identify the quadrant of the angle (165° is in the second quadrant)
2. Find the reference angle by subtracting the angle from 180°: 180° - 165° = 15°
3. Use the identity sin(180° - θ) = sinθ to find sin 165° = sin 15°
4. Calculate sin 15° using known values or a calculator

Step-by-Step Calculation

1. Determine the quadrant: 165° is between 90° and 180°, so it's in the second quadrant.
2. Calculate the reference angle: 180° - 165° = 15°.
3. Apply the sine identity: sin(165°) = sin(15°).
4. Calculate sin(15°). The exact value is (√6 - √2)/4 ≈ 0.2588.

Worked Example

Let's calculate sin(165°) using the reference angle method:

1. Identify the quadrant: 165° is in the second quadrant.
2. Find the reference angle: 180° - 165° = 15°.
3. Use the identity: sin(165°) = sin(15°).
4. Calculate sin(15°):
• Using exact value: sin(15°) = (√6 - √2)/4 ≈ 0.2588
• Using decimal approximation: sin(15°) ≈ 0.2588

The exact value of sin(165°) is (√6 - √2)/4, and the approximate decimal value is 0.2588.

Visualization

The unit circle visualization helps understand the relationship between angles and their sine values. For 165°, the point on the unit circle has coordinates (cos 165°, sin 165°).