Sine Calculator
Enter an angle in degrees or radians to find its sine, sin θ.
Updated
Sine Calculator
Our Sine Calculator finds sin θ for any angle in degrees or radians. Type the angle and the answer appears straight away to 10 decimal places, with the exact value for special angles (such as √3/2 for 60°), the angle in the other unit, the quadrant, and cos θ and tan θ. You can type multiples of π such as π/6, and special angles come out exact, so sin 30° is 0.5 rather than the 0.49999999999999994 many programs show.
Quick answer
The sine of an angle is the opposite side divided by the hypotenuse in a right triangle, or the y-coordinate on the unit circle. For example, sin 30° = 1/2 = 0.5. Enter any angle above and the calculator shows sin θ instantly.
- Formula: sin θ = opposite ÷ hypotenuse
- Key values: sin 0° = 0, sin 30° = 0.5, sin 45° ≈ 0.7071, sin 60° ≈ 0.8660, sin 90° = 1
- Range: always between −1 and 1, repeating every 360° (2π)
- Radians: radians = degrees × π ÷ 180, so 30° = π/6
What Is Sine?
Sine is one of the three main trigonometric functions, with cosine and tangent. In a right triangle it compares the side opposite an angle with the longest side, the hypotenuse:
SOH: sin = opp / hyp
CAH: cos = adj / hyp
TOA: tan = opp / adj
Triangles only cover angles between 0° and 90°. The unit circle extends sine to every angle: walk round a circle of radius 1 from the positive x-axis by the angle θ, and sin θ is the height (y-coordinate) of the point you reach. That is why sine is positive in the top half of the circle, negative in the bottom half, and never larger than 1.
Sine Table of Common Angles
| Angle | Exact | Decimal |
|---|---|---|
| 0° (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 |
| 135° (3π/4) | √2/2 | 0.7071 |
| 150° (5π/6) | 1/2 | 0.5 |
| 180° (π) | 0 | 0 |
| 270° (3π/2) | −1 | −1 |
| 360° (2π) | 0 | 0 |
A handy pattern for 0°, 30°, 45°, 60° and 90°: the sines are √0/2, √1/2, √2/2, √3/2 and √4/2.
The Sine Graph
- Period: the wave repeats every 360° (2π radians).
- Amplitude: it swings between −1 and 1.
- Zeros: sin θ = 0 at 0°, 180°, 360° and every multiple of 180°.
- Odd function: sin(−θ) = −sin θ, so the graph has rotational symmetry about the origin.
Degrees and Radians
A full turn is 360 degrees or 2π radians. One radian is the angle where the arc length equals the radius, about 57.2958°.
degrees = radians × 180 ÷ π
30° = π/6 ≈ 0.5236 rad
1 rad ≈ 57.2958°
Scientific calculators have a DEG/RAD mode, and programming languages always use radians. If an answer looks wrong, check the unit first.
Sine Rules and Identities
- Pythagorean: sin2θ + cos2θ = 1.
- Negative angle: sin(−θ) = −sin θ.
- Supplement: sin(180° − θ) = sin θ, so sin 150° = sin 30° = 0.5.
- Complement: sin(90° − θ) = cos θ.
- Full turn: sin(θ + 360°) = sin θ.
- Double angle: sin 2θ = 2 sin θ cos θ.
- Sum: sin(A + B) = sin A cos B + cos A sin B.
- Law of sines: a ÷ sin A = b ÷ sin B = c ÷ sin C in any triangle.
Worked Examples
Height of a ramp. A 10 m ramp rises at 30°. Its height is the side opposite the angle:
= 10 × 0.5 = 5 m
A ladder against a wall. A 5 m ladder makes a 70° angle with the ground. It reaches 5 × sin 70° ≈ 5 × 0.9397 ≈ 4.70 m up the wall.
Law of sines. In a triangle, the side opposite 60° is 10. The side opposite 40° is:
= 10 × 0.6428 ÷ 0.8660
≈ 7.42
Radian inputs. sin(0.52) ≈ 0.4969, sin(0.79) ≈ 0.7104 and sin(1.57) ≈ 0.9999997, very close to 1 because 1.57 is close to π/2 ≈ 1.5708.
Sine in Excel and Code
| Tool | sin 30° |
|---|---|
| Excel / Google Sheets | =SIN(RADIANS(30)) |
| Python | math.sin(math.radians(30)) |
| JavaScript | Math.sin(30 * Math.PI / 180) |
| Scientific calculator | DEG mode, then sin 30 |
Excel’s SIN and the code functions all expect radians, so =SIN(30) gives −0.9880, the sine of 30 radians. They also return 0.49999999999999994 for sin 30° because π cannot be stored exactly; round the result if you need clean numbers.
Common Mistakes
- Wrong angle mode. sin 30 is 0.5 in degrees but about −0.9880 in radians.
- Reading sin⁻¹ as 1/sin. sin−1(0.5) = 30° (arcsine), while 1 ÷ sin 30° = 2 (cosecant).
- Doubling the angle doubles the sine. It does not: sin 60° ≈ 0.8660, not 2 × 0.5 = 1.
- Splitting a sum. sin(30° + 60°) = sin 90° = 1, not 0.5 + 0.8660.
- Using the wrong side. Sine uses the side opposite the angle, not the side next to it.
Key Terms
- Hypotenuse
- The longest side of a right triangle, opposite the right angle.
- Opposite
- The side across from the angle θ.
- Unit circle
- A circle of radius 1; sin θ is the y-coordinate of the point at angle θ.
- Radian
- The angle whose arc equals the radius; 2π radians make a full turn.
- Period
- The length after which a wave repeats: 360° for sine.
- Arcsine
- The inverse of sine, written sin⁻¹ or asin.
How to Use the Sine Calculator
- Enter the angle. Type the angle θ, for example 30, or press one of the angle buttons (30°, 45°, 60° and 90° in degrees, π/6, π/4, π/3 and π/2 in radians). In radians you can type a decimal such as 0.52 or a multiple of π such as pi/6 or 2π/3.
- Choose degrees or radians. Pick Degrees (°) or Radians (rad). Typing 30° or 0.52 rad also sets the unit for that angle.
- Read and copy the result. sin θ appears as you type, to 10 decimal places, with the exact value where there is one (for example sin(45°) = √2/2 ≈ 0.7071067812). The table below shows the angle in the other unit, the quadrant, cos θ and tan θ. Press Copy to copy the result, or Clear or Esc to start again.
Example: enter 210 in degrees and the result is −0.5, with the exact value −1/2, the note that 210° is in quadrant III with a reference angle of 30°, and 210° = 7π/6 rad.
Who Can Use This Calculator?
Frequently Asked Questions (FAQs)
What is the sine of an angle?
In a right triangle, the sine of an angle θ is the length of the side opposite θ divided by the hypotenuse: sin θ = opposite ÷ hypotenuse. On the unit circle (radius 1), sin θ is the y-coordinate of the point at angle θ, which is how sine is defined for any angle, including negative angles and angles over 90°.
What is sin 30°, sin 45°, sin 60° and sin 90°?
sin 30° = 1/2 = 0.5, sin 45° = √2/2 ≈ 0.7071, sin 60° = √3/2 ≈ 0.8660 and sin 90° = 1. Also sin 0° = 0 and sin 180° = 0. In radians these are sin(π/6), sin(π/4), sin(π/3) and sin(π/2).
Should I use degrees or radians?
Use whatever unit your angle is in. School geometry and everyday angles use degrees; calculus, physics and programming languages use radians. They are linked by 180° = π radians, so radians = degrees × π ÷ 180. Mixing them up is the most common sine mistake: sin(30) in radians is about −0.9880, not 0.5.
Can sine be greater than 1 or negative?
Sine of a real angle always lies between −1 and 1. It is positive for angles between 0° and 180° (quadrants I and II) and negative between 180° and 360° (quadrants III and IV). For example sin 210° = −0.5.
How do you remember sine, cosine and tangent?
Use SOH CAH TOA: Sine = Opposite ÷ Hypotenuse, Cosine = Adjacent ÷ Hypotenuse, Tangent = Opposite ÷ Adjacent.
Is sin⁻¹ the same as 1/sin?
No. sin−1(x) usually means arcsine, the inverse function that gives the angle whose sine is x: sin−1(0.5) = 30°. The reciprocal 1 ÷ sin θ is a different function called cosecant, csc θ: csc 30° = 2.
Why does my calculator or code give 0.49999999999999994 for sin 30°?
Computers store π and 30° in radians only approximately, so Math.sin(Math.PI / 6) in JavaScript and math.sin(math.radians(30)) in Python return 0.49999999999999994. The true value is exactly 0.5. This calculator works in degrees for special angles, so it shows 0.5, and sin 180° as exactly 0 instead of 1.22 × 10−16.
What is the period of the sine function?
360°, or 2π radians. Adding a full turn gives the same value: sin 390° = sin 30° = 0.5. The sine wave repeats every 360°, rising from 0 to 1 at 90°, back to 0 at 180°, down to −1 at 270° and back to 0 at 360°.
What is the law of sines?
In any triangle, each side divided by the sine of its opposite angle gives the same number: a ÷ sin A = b ÷ sin B = c ÷ sin C. If the side opposite a 60° angle is 10, the side opposite a 40° angle is 10 × sin 40° ÷ sin 60° ≈ 7.42.
Is the Sine Calculator free?
Yes. It is completely free, needs no sign-up, and can be used as many times as you like.
Try our Sine Calculator to find sin θ for any angle in degrees or radians instantly.