The value of $\sin\frac{5\pi}{4}$ is $-\frac{\sqrt{2}}{2}$, which is the same as $-\frac{1}{\sqrt{2}} \approx -0.7071$.
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 |
|---|---|---|---|
$\frac{\pi}{4}$ | 45° | I | $\frac{\sqrt{2}}{2}$ |
$\frac{3\pi}{4}$ | 135° | II | $\frac{\sqrt{2}}{2}$ |
$\pi$ | 180° | — | $0$ |
$\frac{5\pi}{4}$ | 225° | III | $-\frac{\sqrt{2}}{2}$ |
$\frac{3\pi}{2}$ | 270° | — | $-1$ |
$\frac{7\pi}{4}$ | 315° | IV | $-\frac{\sqrt{2}}{2}$ |
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 $\frac{180°}{\pi}$ to switch units.
$$\frac{5\pi}{4} \times \frac{180°}{\pi} = \frac{5 \times 180°}{4} = 225°$$
So $\sin\frac{5\pi}{4} = \sin 225°$. If you are more comfortable in degrees, this is your bridge — and it goes both ways, so 225° converts back by multiplying by $\frac{\pi}{180°}$. For a refresher on the conversion factor, see what a radian is.
Final answer: $225°$, ready for the reference-angle step.
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°).
$$\frac{5\pi}{4} - \pi = \frac{5\pi - 4\pi}{4} = \frac{\pi}{4}$$
The reference angle is $\frac{\pi}{4}$ (45°), and $\sin\frac{\pi}{4} = \frac{\sqrt{2}}{2}$.
Now apply the quadrant sign. Quadrant III makes sine negative, so:
$$\sin\frac{5\pi}{4} = -\sin\frac{\pi}{4} = -\frac{\sqrt{2}}{2}$$
Final answer: $-\frac{\sqrt{2}}{2}$.
Method 3: Read the coordinate directly
The point on the unit circle at 225° is $\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$. Sine is the $y$-coordinate, so $\sin\frac{5\pi}{4} = -\frac{\sqrt{2}}{2}$ 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 $\frac{\sqrt{2}}{2}$ from $\sin\frac{\pi}{4}$ is fresh on the page.
Don't do this: Write $\sin\frac{5\pi}{4} = \frac{\sqrt{2}}{2}$ 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 $\pi - \frac{5\pi}{4} = -\frac{\pi}{4}$ and panic at the negative.
The correct way: For a third-quadrant angle the reference angle is (angle − π), so $\frac{5\pi}{4} - \pi = \frac{\pi}{4}$. Match the subtraction to the quadrant.
Mistake 3: Confusing 5π/4 with 5π over 4 of something else
Where it slips in: Reading $\frac{5\pi}{4}$ as $5\pi$ then dividing the result, instead of as a single angle.
Don't do this: Evaluate $\sin 5\pi = 0$ and then divide by 4.
The correct way: $\frac{5\pi}{4}$ 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.
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