Sin Calculator

Q: How do you find the sine of an angle without a calculator?

For the common angles, memorize the exact values: sin(0°)=0, sin(30°)=1/2, sin(45°)=√2/2, sin(60°)=√3/2, sin(90°)=1. For other angles, use a right triangle. Measure the side across from the angle and divide it by the hypotenuse (the longest side). That ratio is the sine.

Q: What is the difference between sin and sin inverse?

Sine takes an angle and gives you a ratio. Sin inverse (also written arcsin or sin⁻¹) does the opposite: it takes a ratio and gives you back the angle. Example: sin(30°) = 0.5, and arcsin(0.5) = 30°. sin⁻¹ does not mean 1 divided by sine. That is the cosecant.

Q: Why can't sine be greater than 1?

In a right triangle, the hypotenuse is always the longest side. So the opposite side divided by the hypotenuse can never be more than 1. On the unit circle, the radius is 1. The y-coordinate of any point on that circle must stay between −1 and 1. Sine is that y-coordinate, so it is trapped in that range.

Q: What is sin(0)?

sin(0°) = 0, and sin(0 radians) = 0 too. At 0°, the point on the unit circle sits at (1, 0). The height is zero, so the sine is zero. Sine is also 0 at 180°, 360°, and every multiple of 180°.

Q: What is sin(90 degrees)?

sin(90°) = 1. This is the largest value sine can reach. At 90°, the point on the unit circle is straight up at (0, 1). The height is 1, so the sine is 1. In radians, 90° is π/2, so sin(π/2) = 1 as well.

Q: What is the sine rule and when do you use it?

The sine rule (law of sines) works for any triangle, not just right triangles: a / sin(A) = b / sin(B) = c / sin(C) Lowercase letters are side lengths. Capital letters are the angles across from them. Use it when you know two angles and one side, or two sides and an angle that is not between them.

Q: How do you convert degrees to radians?

Multiply the degrees by π/180. Example: 60° × π/180 = π/3 ≈ 1.0472 rad. To go the other way, multiply radians by 180/π. So π/4 × 180/π = 45°.

Q: Is sine negative in the third quadrant?

Yes. Sine is negative for any angle between 180° and 360°, which covers Quadrant III and Quadrant IV. That is because the point on the unit circle drops below the x-axis, so its height (y-value) is negative. Sine is positive from 0° to 180°, which is Quadrant I and Quadrant II.

Q: What is the sine of a negative angle?

sin(−θ) = −sin(θ). Flipping the sign of the angle flips the sign of the answer. Example: sin(30°) = 0.5, so sin(−30°) = −0.5. This happens because a negative angle rotates clockwise, putting the point below the x-axis.

Q: What is a reference angle and how do you find it?

A reference angle is the small angle between your angle and the x-axis. It is always between 0° and 90°. To find it: for 0° to 90°, it is the angle itself. For 90° to 180°, use 180° − angle. For 180° to 270°, use angle − 180°. For 270° to 360°, use 360° − angle. The sine of your angle equals the sine of the reference angle, with a plus or minus sign added based on the quadrant.

Sin Calculator

Sine values for commonly used notable angles in degrees and radians
Angle (Degrees)	Angle (Radians)	sin(x)	Exact Value
−90°	−π/2	−1	−1
−45°	−π/4	−0.7071	−√2/2
0°	0	0	0
30°	π/6	0.5	1/2
45°	π/4	0.7071	√2/2
60°	π/3	0.8660	√3/2
90°	π/2	1	1
120°	2π/3	0.8660	√3/2
135°	3π/4	0.7071	√2/2
150°	5π/6	0.5	1/2
180°	π	0	0
270°	3π/2	−1	−1
360°	2π	0	0

Sine values for commonly used notable angles in degrees and radians

Angle (Degrees)

Angle (Radians)

sin(x)

Exact Value

−90°

−π/2

−1

−45°

−π/4

−0.7071

−√2/2

0°

30°

π/6

0.5

1/2

45°

π/4

0.7071

√2/2

60°

π/3

0.8660

√3/2

90°

π/2

120°

2π/3

0.8660

√3/2

135°

3π/4

0.7071

√2/2

150°

5π/6

0.5

1/2

180°

270°

3π/2

−1

360°

2π

Introduction

This Sin Calculator finds the sine of any angle. Type an angle in degrees or radians, and the sine value appears. You can also type pi expressions like π/4, 2π, or 3*π/2.

Sine is one of the three main trig functions, along with cosine and tangent. On the unit circle, sin(x) is the height (the y-coordinate) of the point at that angle. That is why sine always stays between −1 and 1.

Along with the number itself, the calculator shows:

The sine value rounded to 8 decimal places
The exact value, like 1/2 or √2/2, when one exists
Your angle in both degrees and radians
A unit circle picture with your angle drawn on it
A sine wave graph with your point marked
Step-by-step work that shows how the answer was found
A table of sine values for common angles

It works well for homework, test review, geometry, physics, and any job that uses angles and waves. Enter your angle and read the answer.

How to use our Sin Calculator

Type in your angle and pick its unit. The calculator shows the sine value, the exact form when one exists, your angle in both degrees and radians, a unit circle, a sine curve, and the full step-by-step math.

Angle: Enter the angle you want the sine of. You can type a plain number like 45 or 1.5708, or a pi expression like π/4, 2π, or 3*π/2. Tap the π button to drop a pi symbol into the box.

Angle Unit: Choose Degrees (°) or Radians (rad) to tell the calculator how to read your number. Degrees is the default. Either way, both forms of the angle show up in the results.

Calculate: Click this button to run the math and update the sin value, unit circle, chart, and steps. Results also refresh as you type.

Reset: Click this to clear your work and go back to the starting angle of 30 degrees.

What Is Sine (sin)?

Sine is one of the three main trig functions, along with cosine and tangent. In a right triangle, the sine of an angle is the length of the side across from that angle divided by the longest side (the hypotenuse).

sin(θ) = opposite ÷ hypotenuse

Sine on the Unit Circle

A unit circle is a circle with a radius of 1. Draw an angle from the center, and the point where it hits the circle has an x and a y value. The y value is the sine of that angle. This is why sine works for any angle, not just the ones inside a triangle. That includes angles bigger than 90° and negative angles.

Degrees and Radians

Angles can be measured two ways. Degrees split a full circle into 360 parts. Radians use π, where a full circle is 2π. To switch between them:

Degrees to radians: multiply by π/180
Radians to degrees: multiply by 180/π

So 30° = π/6, 90° = π/2, and 180° = π.

Key Facts About Sine

The answer is always between −1 and 1. Nothing else is possible.
Sine repeats every 360° (2π). That means sin(30°) and sin(390°) give the same answer.
Sine is positive for angles in Quadrant I and II (0° to 180°), and negative in Quadrant III and IV (180° to 360°).
sin(−θ) = −sin(θ). Flipping the angle flips the sign.
The graph of sine is a smooth wave that rolls up and down forever.

Common Sine Values Worth Knowing

A few angles have clean, exact answers you can memorize:

sin(0°) = 0
sin(30°) = 1/2
sin(45°) = √2/2 ≈ 0.7071
sin(60°) = √3/2 ≈ 0.8660
sin(90°) = 1

Where Sine Is Used

Sine shows up any time something repeats or any time you need a missing side or angle. It is used to find heights and distances in building and surveying, to describe sound waves and light waves, to model electricity in AC circuits, to track tides and seasons, and to move objects smoothly in games and animation.

Formulas used

Sine of an angle (radians)

y = \sin(\theta_{\text{rad}})

Degrees to radians conversion

\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}

Radians to degrees conversion

\theta_{\text{deg}} = \theta_{\text{rad}} \times \frac{180}{\pi}

Coterminal angle (normalized to 0°–360°)

\theta_n = \theta_{\text{deg}} \bmod 360^\circ

Reference angle by quadrant

\theta_{\text{ref}} = \begin{cases} \theta_n & 0^\circ \le \theta_n \le 90^\circ \\ 180^\circ - \theta_n & 90^\circ < \theta_n \le 180^\circ \\ \theta_n - 180^\circ & 180^\circ < \theta_n \le 270^\circ \\ 360^\circ - \theta_n & 270^\circ < \theta_n < 360^\circ \end{cases}

Sine from reference angle with quadrant sign

\sin(\theta_n) = \pm\sin(\theta_{\text{ref}})

Unit circle point coordinates

(x, y) = (\cos\theta,\ \sin\theta)

Radian measure as a multiple of pi

\theta_{\text{rad}} = \frac{\theta_{\text{deg}}}{180}\pi

Frequently asked questions

How do you find the sine of an angle without a calculator?

For the common angles, memorize the exact values: sin(0°)=0, sin(30°)=1/2, sin(45°)=√2/2, sin(60°)=√3/2, sin(90°)=1.

For other angles, use a right triangle. Measure the side across from the angle and divide it by the hypotenuse (the longest side). That ratio is the sine.

What is the difference between sin and sin inverse?

Sine takes an angle and gives you a ratio. Sin inverse (also written arcsin or sin⁻¹) does the opposite: it takes a ratio and gives you back the angle.

Example: sin(30°) = 0.5, and arcsin(0.5) = 30°.

sin⁻¹ does not mean 1 divided by sine. That is the cosecant.

Why can't sine be greater than 1?

In a right triangle, the hypotenuse is always the longest side. So the opposite side divided by the hypotenuse can never be more than 1.

On the unit circle, the radius is 1. The y-coordinate of any point on that circle must stay between −1 and 1. Sine is that y-coordinate, so it is trapped in that range.

What is sin(0)?

sin(0°) = 0, and sin(0 radians) = 0 too.

At 0°, the point on the unit circle sits at (1, 0). The height is zero, so the sine is zero.

Sine is also 0 at 180°, 360°, and every multiple of 180°.

What is sin(90 degrees)?

sin(90°) = 1. This is the largest value sine can reach.

At 90°, the point on the unit circle is straight up at (0, 1). The height is 1, so the sine is 1.

In radians, 90° is π/2, so sin(π/2) = 1 as well.

What is the sine rule and when do you use it?

The sine rule (law of sines) works for any triangle, not just right triangles:

a / sin(A) = b / sin(B) = c / sin(C)

Lowercase letters are side lengths. Capital letters are the angles across from them.

Use it when you know two angles and one side, or two sides and an angle that is not between them.

How do you convert degrees to radians?

Multiply the degrees by π/180.

Example: 60° × π/180 = π/3 ≈ 1.0472 rad.

To go the other way, multiply radians by 180/π. So π/4 × 180/π = 45°.

Is sine negative in the third quadrant?

Yes. Sine is negative for any angle between 180° and 360°, which covers Quadrant III and Quadrant IV.

That is because the point on the unit circle drops below the x-axis, so its height (y-value) is negative.

Sine is positive from 0° to 180°, which is Quadrant I and Quadrant II.

What is the sine of a negative angle?

sin(−θ) = −sin(θ). Flipping the sign of the angle flips the sign of the answer.

Example: sin(30°) = 0.5, so sin(−30°) = −0.5.

This happens because a negative angle rotates clockwise, putting the point below the x-axis.

What is a reference angle and how do you find it?

A reference angle is the small angle between your angle and the x-axis. It is always between 0° and 90°.

To find it: for 0° to 90°, it is the angle itself. For 90° to 180°, use 180° − angle. For 180° to 270°, use angle − 180°. For 270° to 360°, use 360° − angle.

The sine of your angle equals the sine of the reference angle, with a plus or minus sign added based on the quadrant.

Why does sin(30) and sin(390) give the same answer?

Sine repeats every 360°. Going around the circle once brings you back to the same spot.

390° − 360° = 30°, so both land on the same point on the unit circle. Both equal 0.5.

This repeat is called the period. In radians, the period is 2π.

What is the exact value of sin(45 degrees)?

sin(45°) = √2/2, which is about 0.70710678.

You may also see it written as 1/√2. Both are the same number.

At 45°, a right triangle has two equal legs, so the height and width match. In radians, 45° is π/4.

How are sine and cosine related?

Cosine is sine shifted by 90°. The rule is sin(θ) = cos(90° − θ).

Example: sin(30°) = cos(60°) = 0.5.

On the unit circle, sine is the y-coordinate and cosine is the x-coordinate of the same point. They also follow sin²(θ) + cos²(θ) = 1.

What is sine used for in real life?

Sine helps find heights and distances you cannot measure directly, like the height of a tree or building from a known angle and distance.

It also describes anything that repeats in a wave: sound, light, radio signals, AC electricity, tides, and daylight hours through the year.

Engineers, builders, surveyors, pilots, and game designers all use it.

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