What is the sin of my angle?
Type an angle in degrees or radians. The sin calculator shows sin θ, its exact value when the angle is a multiple of 15°, csc θ, the reference angle and the quadrant.
- sin θ
- 0.5
sin(30°) = 0.5.
- Exact value
- sin 30° = 1/2
- csc θ
- 2
- Reference angle
- 30°
- Quadrant
- Quadrant I
sin θ: 0.5. sin(30°) = 0.5.
How to calculate
Finds sin θ of any angle in degrees or radians, with the exact value at multiples of 15°, csc θ, and the reference angle.
Example with the default inputs (Angle θ 30 °): sin(30°) = 0.5.
Method: sin θ is the y value of the point at angle θ on the unit circle. At a multiple of 15° the exact value comes from the special angles and the quadrant sign; otherwise sin θ is computed in 64-bit floating point. csc θ = 1 ÷ sin θ.
- The angle can be typed in degrees or radians, from −1,000,000 to 1,000,000 radians, and can be negative or more than 360°.
- An angle within one part in 10¹² of a multiple of 15° counts as that multiple, so sin 180° is exactly 0.
- csc θ shows nothing where sin θ = 0 (0°, 180°, 360°, …).
Worked examples
Each example is checked against the calculator on every build.
- Angle θ 30° gives sin θ 0.5, Exact value sin 30° = 1/2, csc θ 2, Reference angle 30°, Quadrant Quadrant I.Source: OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (special angles: sin 30° = 1/2, sin 45° = √2/2). https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle
- Angle θ 225° gives sin θ -0.707107, Exact value sin 225° = -√2/2, Quadrant Quadrant III, Reference angle 45°.Source: OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (special angles: sin 30° = 1/2, sin 45° = √2/2). https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle
- Angle θ 180° gives sin θ 0, Exact value sin 180° = 0, Quadrant On the negative x-axis.Source: OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (special angles: sin 30° = 1/2, sin 45° = √2/2). https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle
- Angle θ 57.3° gives sin θ 0.841471, csc θ 1.188395.Source: NIST DLMF §4.14 (sine of a real angle), https://dlmf.nist.gov/4.14
- Angle θ 200° gives sin θ -0.34202, Quadrant Quadrant III.Source: NIST DLMF §4.14, https://dlmf.nist.gov/4.14
How it works
Draw the angle θ from the positive x-axis, turning counterclockwise for a positive angle. It meets the unit circle (radius 1) at the point (cos θ, sin θ). The sine is the y value:
- sin θ = y, and in a right triangle, sin θ = opposite ÷ hypotenuse.
- csc θ = 1 ÷ sin θ, undefined where sin θ = 0.
- Reference angle: the acute angle between θ and the x-axis, from 0° to 90°.
- Quadrant: I (0° to 90°), II (90° to 180°), III (180° to 270°) or IV (270° to 360°), after whole turns are taken off. An angle that ends on an axis shows the axis instead.
Exact values. When θ is a multiple of 15° (π/12 radians), the calculator shows the exact value. It takes the reference angle's sine from this table and gives it the sign of the quadrant (positive in I and II, negative in III and IV):
| Reference angle | 0° | 15° | 30° | 45° | 60° | 75° | 90° |
|---|---|---|---|---|---|---|---|
| sin | 0 | (√6 − √2)/4 | 1/2 | √2/2 | √3/2 | (√6 + √2)/4 | 1 |
An angle within one part in 10¹² (relative, and at least 10⁻¹² radians) of a multiple of 15° counts as that multiple. So sin 180° is exactly 0, not the 1.2 × 10⁻¹⁶ that a rounded π gives. For any other angle, sin θ is computed in 64-bit floating point.
Output format. sin θ and csc θ show as decimal numbers. The exact value shows as text, such as sin 30° = 1/2 (- for a negative value). The reference angle shows in degrees.
Assumptions
- The angle is typed in degrees or radians, from −1,000,000 to 1,000,000 radians. Negative angles turn clockwise.
- csc θ shows nothing at 0°, 180°, 360° and the other angles where sin θ = 0.
Worked examples by hand
θ = 30°. 30° is in quadrant I, so sin 30° = 1/2 = 0.5 and csc 30° = 2. The reference angle is 30°.
θ = 225°. 225° − 180° = 45°, so the reference angle is 45° and the angle is in quadrant III, where the sine is negative: sin 225° = −√2/2 ≈ −0.7071067811865476.
θ = 180°. The point is (−1, 0), so sin 180° = 0, on the negative x-axis. csc 180° is undefined.
θ = 1 radian. 1 rad ≈ 57.2958°, in quadrant I. sin 1 ≈ 0.8414709848078965, and csc 1 = 1 ÷ 0.8414709848078965 ≈ 1.1883951057781212.
θ = 200°. 200° is in quadrant III (reference angle 20°), so the sine is negative: sin 200° ≈ −0.34202014332566866.
Other questions people ask
What is the sine of an angle?
Draw the angle θ from the positive x-axis on a circle of radius 1 centred at the origin. The point where it meets the circle is (cos θ, sin θ), so sin θ is its y value. In a right triangle with acute angle θ, sin θ = opposite side ÷ hypotenuse.
What is sin 30°?
sin 30° = 1/2 = 0.5 exactly. The other common values are sin 45° = √2/2 ≈ 0.7071, sin 60° = √3/2 ≈ 0.8660 and sin 90° = 1.
Should I use degrees or radians?
Use the unit the angle is given in. A full turn is 360° or 2π radians, so 1 radian is about 57.2958°. Typing 30 with the unit set to radians gives sin 30 rad ≈ −0.9880, not 0.5.
Can sine be negative?
Yes. sin θ is negative when the point on the unit circle is below the x-axis: for angles between 180° and 360° (quadrants III and IV). sin 225° = −√2/2.
What is the largest value of sin θ?
sin θ is always between −1 and 1. It is 1 at 90° (plus any whole number of turns) and −1 at 270°.
What is csc θ?
The cosecant, csc θ = 1 ÷ sin θ. It is undefined where sin θ = 0 (0°, 180°, 360°), so the calculator leaves it empty there.