Math
Sine Calculator
Sine is the ratio opposite / hypotenuse in a right triangle and the y-coordinate on the unit circle.
This sine calculator evaluates sin θ in degrees or radians with exact answers such as sin 30° = 1/2 and sin 45° = √2/2, solves sin θ = x for every angle, and finds a missing right-triangle side or angle.
Type a number, a fraction like 180/7, a π expression like 5π/6, or degrees-minutes-seconds like 30° 15′ 30″. Negative and large angles are fine.
Try an example
Result
sine result
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- Angle in degrees
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- Angle in radians
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- Quadrant
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- Reference angle
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- Reciprocal (csc)
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- Cofunction / ratio
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More details (1 more)
- More
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
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Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What sine measures
In a right triangle, the sine of an acute angle compares the side across from it with the longest side. On the unit circle the same number is the height (y-coordinate) of the point at angle θ, which is how sine extends to angles beyond 90° and to negative angles.
Exact sine values
| θ | sin θ | Decimal |
|---|---|---|
| 0° | 0 | 0 |
| 30° (π/6) | 1/2 | 0.5 |
| 45° (π/4) | √2/2 | 0.7071 |
| 60° (π/3) | √3/2 | 0.8660 |
| 90° (π/2) | 1 | 1 |
| 120° (2π/3) | √3/2 | 0.8660 |
| 150° (5π/6) | 1/2 | 0.5 |
| 180° (π) | 0 | 0 |
| 270° (3π/2) | -1 | -1 |
Four ways to use this calculator
| Mode | You enter | You get |
|---|---|---|
| Find sin θ | An angle in any unit | sin θ exact and decimal, csc θ, cos of the complement, quadrant, reference angle |
| Find θ (arcsin) | A value between -1 and 1 | Principal angle, all angles in [0°, 360°), general solution |
| Right-triangle side | An acute angle and the opposite or hypotenuse | The other of the two, plus the third side and other angle |
| Right-triangle angle | Opposite and hypotenuse | The angle, adjacent side, and other angle |
How to read the answer
- sin θ is positive in One of the four regions the x- and y-axes cut the coordinate plane into. I and II (0° to 180°) and negative in Quadrants III and IV.
- Its value never leaves the interval -1 to 1, reaching 1 at 90° and -1 at 270°.
- A result like 1/2 (0.5) shows the The answer kept as a fraction or radical instead of a rounded decimal. first and the rounded decimal second.
- The cofunction line shows cos(90° − θ), which always equals sin θ.
Common mistakes and edge cases
- sin 30 in An angle unit where a full turn is 2π instead of 360 degrees. mode is -0.988, not 0.5. Match the unit selector to your problem.
- sin θ = 1.2 has no solution: the calculator reports the domain error instead of a wrong angle.
- In the triangle modes the angle must be acute (between 0° and 90°); otherwise it is not a right triangle's other angle.
- The opposite side is the one that does not touch the angle. The The longest side of a right triangle, always opposite the right angle. is always across from the right angle.
Worked examples
sin 30°
sine result
1/2 (0.5)
sin of a Quadrant III angle
sin 210° = -1/2
sine result
-1/2 (-0.5)
sin in radians
sin(π/4) = √2/2
sine result
√2/2 (0.707107)
sin 0°
zero case
sine result
0
Solve sin θ = 0.5
sine result
30° = π/6
Impossible sine
sin θ = 2 has no solution
sine result
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Find the opposite side
θ = 30°, hypotenuse = 10
sine result
Opposite side = 5
Find the hypotenuse
θ = 30°, opposite = 4
sine result
Hypotenuse = 8
Angle from two sides
opposite 3, hypotenuse 5
sine result
θ = 36.8699°
Opposite longer than hypotenuse
sine result
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Frequently asked questions
What is sin 30 degrees?+
sin 30° = 1/2 exactly. In a 30-60-90 triangle the side opposite the 30° angle is half the hypotenuse.
How do I calculate sine without a calculator?+
For special angles use the table: 0, 1/2, √2/2, √3/2, 1 for 0°, 30°, 45°, 60°, 90°. For other angles, reduce to the reference angle, look up or estimate, and apply the quadrant sign.
Why is sin 150° the same as sin 30°?+
150° and 30° share the reference angle 30°, and sine is positive in both Quadrant I and Quadrant II. sin(180° − θ) = sin θ for every θ.
Can sine be greater than 1?+
No. Sine is a y-coordinate on a circle of radius 1, so it stays between -1 and 1. If sin θ = 1.2 appears in a problem, something upstream is wrong.
How do I find a side with sine?+
Use sin θ = opposite / hypotenuse. If you know the hypotenuse, multiply: opposite = hypotenuse × sin θ. If you know the opposite side, divide: hypotenuse = opposite / sin θ.
What is the inverse of sine?+
Arcsine (sin⁻¹). It returns the angle between -90° and 90° whose sine is the given value. The calculator also lists the second angle in one full turn, 180° − θ.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026