Evaluate Sin 135 Degrees Without Using A Calculator

How to Evaluate sin 135 Degrees Without a Calculator

An Interactive Explainer for Understanding Trigonometric Functions

Interactive Unit Circle Explainer

Angle

Enter an angle in degrees to see how its sine value is determined.

sin(135°) = √2 / 2 ≈ 0.7071

Quadrant

Reference Angle

45°

Sign of Sine

Positive (+)

What is ‘evaluate sin 135 degrees without using a calculator’?

To “evaluate sin 135 degrees without using a calculator” means finding the exact value of the sine function for an angle of 135° using principles of trigonometry. This process relies on understanding the unit circle, which is a circle with a radius of 1 centered at the origin of a coordinate plane. The sine of an angle is the y-coordinate of the point where the terminal side of that angle intersects the unit circle. For 135°, this means we are looking for a value in the second quadrant.

The key is to use a “reference angle”—the acute angle that the terminal side of 135° makes with the x-axis. By knowing the sine of this simpler reference angle and the sign conventions for each quadrant, we can determine the final value. It’s a fundamental skill in trigonometry that builds a deeper understanding than simply typing numbers into a device.

The Formula and Explanation to Evaluate sin(135°)

There isn’t a single “formula” for this, but rather a repeatable four-step method rooted in the properties of the unit circle. The value of sin(135°) is positive because it lies in the second quadrant where y-values are positive.

Locate the Angle: Start at the positive x-axis and rotate counter-clockwise. An angle of 135° passes 90° and ends in the second quadrant.
Find the Reference Angle (α): The reference angle is the shortest angle back to the x-axis. For an angle θ in Quadrant II, the formula is α = 180° - θ. In this case, α = 180° – 135° = 45°.
Determine the Sign: In the second quadrant, y-coordinates are positive, so the sine function is positive. A helpful mnemonic is “All Students Take Calculus,” where “Students” (for Quadrant II) tells us Sine is positive.
Evaluate the Reference Angle: Find the sine of the reference angle. The value for sin(45°) is a well-known trigonometric value derived from a 45-45-90 triangle. The exact value is sin(45°) = √2 / 2.

Combining these, we get: sin(135°) = +sin(45°) = √2 / 2.

Trigonometric Sign Conventions by Quadrant
Quadrant	Angle Range	Sine (sin)	Cosine (cos)	Tangent (tan)
I	0° – 90°	+	+	+
II	90° – 180°	+	–	–
III	180° – 270°	–	–	+
IV	270° – 360°	–	+	–

Practical Examples

Example 1: Evaluate sin(135°)

Input Angle: 135°
Step 1 (Location): Quadrant II.
Step 2 (Reference Angle): 180° – 135° = 45°.
Step 3 (Sign): Sine is positive in Quadrant II.
Step 4 (Evaluation): sin(45°) = √2 / 2.
Result: sin(135°) = √2 / 2 ≈ 0.7071.

Example 2: Evaluate sin(210°)

Input Angle: 210°
Step 1 (Location): Quadrant III.
Step 2 (Reference Angle): 210° – 180° = 30°.
Step 3 (Sign): Sine is negative in Quadrant III.
Step 4 (Evaluation): sin(30°) = 1 / 2.
Result: sin(210°) = -1 / 2 = -0.5.

How to Use This ‘evaluate sin 135 degrees without using a calculator’ Tool

This interactive tool helps you visualize the process of finding the sine of any angle.

Enter an Angle: Type any angle in degrees into the input field. The calculator defaults to 135° to demonstrate the primary topic.
Observe the Chart: The unit circle diagram will automatically update. The blue arc shows the angle you entered, and the orange arc highlights the calculated reference angle. The red line is the terminal side of the angle, and the green dashed line represents the sine value (the y-coordinate).
Review the Results: The boxes below the chart instantly show you the Quadrant, the Reference Angle, the sign of the sine function for that quadrant, and the final calculated value, both in its exact (radical) form and as a decimal approximation. To better understand reference angles, you can use a dedicated reference angle calculator.

Key Factors That Affect the Result

The Angle’s Quadrant: This is the most crucial factor, as it determines the sign (+ or -) of the result.
The Reference Angle: The entire calculation depends on correctly identifying this acute angle.
Special Triangles (45-45-90 & 30-60-90): The exact values for angles like 30°, 45°, and 60° come from the fixed side ratios in these special right triangles.
Unit (Degrees vs. Radians): While this calculator uses degrees, all angles have a corresponding radian measure (e.g., 135° = 3π/4 radians). The calculation method is the same. You can convert between them using a radian to degree converter.
Trigonometric Identity: The core principle that sin(θ) = y-coordinate on the unit circle is the foundation of this entire process.
Coterminal Angles: Adding or subtracting 360° from an angle gives a coterminal angle with the same trigonometric values. For example, sin(135°) is the same as sin(135° + 360°) = sin(495°).

FAQ about Evaluating Sine

What is the exact value of sin 135 degrees?

The exact value is √2 / 2. This is derived from its reference angle of 45°, for which the sine is √2 / 2.

Why is sin(135) positive?

Because 135° is in the second quadrant of the unit circle. In this quadrant, all y-coordinates are positive, and the sine of an angle corresponds to the y-coordinate.

How do you find the reference angle for 135 degrees?

For any angle in the second quadrant, you subtract the angle from 180°. So, the reference angle is 180° – 135° = 45°.

What is sin(135) in radians?

First, convert 135 degrees to radians: 135 * (π / 180) = 3π / 4. Therefore, sin(135°) is the same as sin(3π/4).

Can you evaluate cos(135) the same way?

Yes. The process is identical, but the sign is different. Cosine corresponds to the x-coordinate, which is negative in Quadrant II. So, cos(135°) = -cos(45°) = -√2 / 2. To explore this further, see our cosine calculator.

What is the value of sin(-135) degrees?

A -135° angle is in Quadrant III. The reference angle is 180° – 135° = 45°. In Quadrant III, sine is negative. Therefore, sin(-135°) = -sin(45°) = -√2 / 2.

What’s the difference between sin(135) and sin(45)?

They have the same absolute value (√2 / 2), but their angles are in different quadrants. sin(45°) is in Quadrant I, while sin(135°) is in Quadrant II. This distinction is critical for solving trigonometric equations.

What are the main trigonometry formulas I should know?

Understanding the unit circle, SOHCAHTOA, and the rules for reference angles are most important for this type of problem. For a full overview, you can check our guide on trigonometry formulas.

Related Tools and Internal Resources

To deepen your understanding of trigonometry, explore these related tools and guides:

Unit Circle Calculator: A comprehensive tool for exploring all values on the unit circle.
Reference Angle Calculator: Quickly find the reference angle for any given angle.
Cosine Calculator: Learn how to evaluate the cosine function using the same principles.
Tangent Calculator: Understand the tangent function, which is the ratio of sine to cosine.
Radian to Degree Converter: Easily switch between the two most common units for measuring angles.
Trigonometry Formulas: A complete list of important trigonometric identities and formulas.