Book A Free Math Class # Sin 5pi/4 — Exact Value, Unit Circle, Methods [Trigonometry](/content/tag/trigonometry/index.html) 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](/content/authors/bhanzu-team/index.html) 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. | Angle (radians) | Angle (degrees) | Quadrant | sin value | | --- | --- | --- | --- | | π/4 | 45° | I | √2/2 | | 3π/4 | 135° | II | √2/2 | | π | 180° | — | 0 | | 5π/4 | 225° | III | −√2/2 | | 3π/2 | 270° | — | −1 | | 7π/4 | 315° | IV | −√2/2 | ## What Sine of an Angle Means Sine is one of the three core [trigonometric ratios](/content/math/trigonometry/trigonometric-ratios/index.html). 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](/content/math/trigonometry/what-is-a-radian/index.html). **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](/content/math/tutor/trigonometry/index.html) and general [math classes online](/content/math/classes/index.html) 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.