Evaluate Sin 135 Degrees Without Using A Calculator

Calculating the sine of 135 degrees without a calculator requires understanding of trigonometric identities and reference angles. This guide explains the methods and provides an interactive calculator to verify your results.

How to calculate sin 135° without a calculator

There are two primary methods to evaluate sin 135° without a calculator: using trigonometric identities and using reference angles. Both methods rely on the unit circle and fundamental trigonometric relationships.

The sine function is periodic with a period of 360°, meaning sin(θ) = sin(θ + 360°n) for any integer n. This property allows us to evaluate angles beyond the first rotation.

135° is located in the second quadrant of the unit circle, where sine values are positive. This is important because the sign of the sine function changes in different quadrants.

Using trigonometric identities

The sine of 135° can be expressed using the angle addition formula:

sin(135°) = sin(180° - 45°) = sin(180°)cos(45°) - cos(180°)sin(45°)

Since sin(180°) = 0 and cos(180°) = -1, this simplifies to:

sin(135°) = 0·cos(45°) - (-1)·sin(45°) = sin(45°)

We know that sin(45°) = √2/2 ≈ 0.7071. Therefore, sin(135°) = √2/2.

Using reference angles

The reference angle for 135° is calculated as 180° - 135° = 45°. Since 135° is in the second quadrant where sine is positive, we have:

sin(135°) = sin(45°) = √2/2

This method is simpler and directly uses the known value of sin(45°).

Worked example

Let's verify the calculation using both methods:

Using angle addition: sin(135°) = sin(180° - 45°) = sin(45°) = √2/2 ≈ 0.7071
Using reference angle: sin(135°) = sin(45°) = √2/2 ≈ 0.7071

Both methods yield the same result, confirming that sin(135°) = √2/2.

Frequently Asked Questions

Why is sin(135°) positive?

135° is in the second quadrant of the unit circle where sine values are positive. This is because the y-coordinate is positive in this quadrant.

Can I use the calculator to verify this result?

Yes, the interactive calculator on this page can help you verify the result by entering 135 degrees and seeing that the sine value matches √2/2 ≈ 0.7071.

What is the reference angle for 135°?

The reference angle for 135° is 45°, calculated as 180° - 135° = 45°. This is useful for evaluating trigonometric functions in the second quadrant.