Book A Free Math Class

Sin 5pi/4 — Exact Value, Unit Circle, Methods

Trigonometry

TL;DR

Sin 5pi/4 is −√2/2 (about −0.7071), because the angle 5π/4 lands at 225° in the third quadrant where sine is negative. This article shows the value through the unit circle, the radian-to-degree conversion, and the π/4 reference-angle method.

BT

Bhanzu Team Last updated on July 16, 20265 min read

The value of sin⁡5π4 is −√2/2, which is the same as −1/√2 ≈ −0.7071.

Quick Answer:

Result: sin(5π/4) = −√2/2 ≈ −0.7071

Notation: exact surd form −√2/2 (equivalently −1/√2)

Method shown: radian → degree conversion + reference angle on the unit circle

Degree equivalent: sin 225°

Sign: negative (third quadrant)

Quick Reference Table for Sin 5pi/4

A few neighbouring angles, written in radians and degrees, with their sine values for comparison.

What Sine of an Angle Means

Sine is one of the three core trigonometric ratios. On the unit circle — a circle of radius 1 centred at the origin — the sine of an angle is the y-coordinate of the point where the angle's terminal side meets the circle.

A quadrant is one of the four regions the x- and y-axes cut the plane into, numbered I to IV anticlockwise from the top-right. Sine is positive in quadrants I and II (where y>0) and negative in III and IV (where y<0). The angle 5π/4 sits in the third quadrant, so its sine is below zero before you compute a single number.

Methods to Find Sin 5pi/4

How do you find the value of sin 5pi/4? Three routes reach the same answer; pick the one that fits how the angle is given to you.

Method 1: Convert radians to degrees first

Multiply by 180°/π to switch units.

5π/4 × 180°/π = 5 × 180°/4 = 225°.

So sin⁡5π4 = sin⁡225°. If you are more comfortable in degrees, this is your bridge — and it goes both ways, so 225° converts back by multiplying by π/180°. For a refresher on the conversion factor, see what a radian is.

Final answer: 225°.

Method 2: Reference angle on the unit circle

The reference angle is the acute angle between the terminal side and the x-axis. For a third-quadrant angle you subtract π (or 180°).

5π/4 − π = π/4.

The reference angle is π/4 (45°), and sin⁡π4 = √2/2.

Now apply the quadrant sign. Quadrant III makes sine negative, so:

sin⁡5π4 = −sin⁡π4 = −√2/2.

Final answer: −√2/2.

Method 3: Read the coordinate directly

The point on the unit circle at 225° is (−√2/2, −√2/2). Sine is the y-coordinate, so sin⁡5π4 = −√2/2 by inspection. This is the fastest check once the unit-circle picture is in your head.

The value is identical for the degree form, so the radian page and the degree page describe the same point on the circle — only the label on the angle changes.

Common Mistakes of Sin 5pi/4

Mistake 1: Forgetting the negative sign

Where it slips in: Right after finding the reference angle, when the clean √2/2 from sin⁡π4 is fresh on the page.

Don't do this: Write sin⁡5π4 = √2/2 because the reference value is positive.

The correct way: Apply the quadrant sign as a separate step — the reference angle gives the size, the quadrant gives the sign, and quadrant III makes sine negative.

Mistake 2: Subtracting the wrong base for the reference angle

Where it slips in: Treating every angle like a second-quadrant one and subtracting from π.

Don't do this: Compute π − 5π/4 = −π/4 and panic at the negative.

The correct way: For a third-quadrant angle the reference angle is (angle − π), so 5π/4 − π = π/4. Match the subtraction to the quadrant.

Mistake 3: Confusing 5π/4 with 5π over 4 of something else

Where it slips in: Reading 5π/4 as 5π and then dividing the result, instead of as a single angle.

Don't do this: Evaluate sin⁡5π = 0 and then divide by 4.

The correct way: 5π/4 is one angle, equal to 225°. Convert it whole before taking the sine.

Sin 5pi/4 is one of the standard third-quadrant values worth knowing cold; to work through more of them with a live teacher, Bhanzu's trigonometry tutor and general math classes online cover the unit circle from the ground up.

Frequently Asked Questions

Is sin 5pi/4 positive or negative?

Negative. The angle is in the third quadrant, where the y-coordinate — and therefore sine — is below zero.

What is sin 5pi/4 in decimal form?

About −0.7071. The exact form −√2/2 is preferred for written work because it never rounds.

Why is the reference angle π/4?

Because 5π/4 is exactly π/4 past π (180°). The terminal side makes a 45° angle with the negative x-axis, and that acute angle is the reference angle.

How is sin 5pi/4 related to cos 5pi/4?

At 225° both coordinates are equal and negative, so sin⁡5π4 = cos⁡5π4 = −√2/2. That is why tan⁡5π4 = 1.

Does sin 5pi/4 equal sin 225°?

Yes. They are the same angle in two notations — 5π/4 radians is exactly 225°, so the sine is identical: −√2/2.