How to Find Sin 135 Minus Sin 270 Without Calculator

Introduction

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°).

For angles outside the first period, we can use reference angles to find their equivalent within the first period. A reference angle is the smallest angle that the terminal side of a given angle makes with the x-axis.

Trigonometric Identities

Key trigonometric identities that help simplify calculations:

• sin(180° - θ) = sinθ
• sin(180° + θ) = -sinθ
• sin(360° - θ) = -sinθ

These identities help convert angles to their equivalent forms within the first period.

Reference Angles

Reference angles help simplify calculations for angles outside the first period. The reference angle for an angle θ is the smallest angle between the terminal side of θ and the x-axis.

For example, the reference angle for 135° is 45° (180° - 135°), and for 270° is 90° (270° - 180°).

Calculation Steps

1. Convert 135° to its reference angle: 135° - 180° = -45° → absolute value is 45°.
2. Determine the sign of sin(135°): Since 135° is in the second quadrant where sine is positive, sin(135°) = sin(45°).
3. Convert 270° to its reference angle: 270° - 180° = 90°.
4. Determine the sign of sin(270°): Since 270° is on the negative y-axis where sine is negative, sin(270°) = -sin(90°).
5. Calculate the values: sin(45°) = √2/2 ≈ 0.7071, sin(90°) = 1.
6. Compute the final result: sin(135°) - sin(270°) = (√2/2) - (-1) = √2/2 + 1 ≈ 1.7071.

Example

Let's calculate sin(135°) - sin(270°):

1. sin(135°) = sin(180° - 45°) = sin(45°) = √2/2 ≈ 0.7071
2. sin(270°) = sin(180° + 90°) = -sin(90°) = -1
3. Result: 0.7071 - (-1) = 1.7071