Sin Calculator

Free Online Sin Calculator — Solve sin(θ) Instantly

This sin calculator (sine calculator) is a complete trigonometry tool that solves sine problems in every direction you're likely to face — plug in an angle to get sin(θ), plug in the opposite side and hypotenuse of a right triangle to get the angle, or plug in a decimal sine value to solve an equation like sin(θ) = 0.5 for θ. Every calculation is backed by a to-scale diagram and a full step-by-step solution, so you can see exactly how the sine value or angle was derived instead of just getting a bare number.

Whether you're a student solving trigonometry homework, checking a physics or engineering problem involving angles and forces, or just need a quick sine, cosine, and tangent value for an angle in degrees or radians, this calculator covers the full sin, cos, tan relationship in one place.

What Is Sine (sin) in Trigonometry?

Sine is one of the three primary trigonometric ratios, alongside cosine and tangent. For any angle θ measured in a right triangle, sine is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse — the longest side, opposite the right angle. This simple ratio, written sin(θ) = Opposite ÷ Hypotenuse, is the foundation of the SOH-CAH-TOA mnemonic taught in every trigonometry class:

• SOH — Sine = Opposite ÷ Hypotenuse
• CAH — Cosine = Adjacent ÷ Hypotenuse
• TOA — Tangent = Opposite ÷ Adjacent

The Sine Formula and the Unit Circle

Beyond right triangles, sine is extended to any angle — even angles greater than 90° or negative angles — using the unit circle, a circle of radius 1 centered at the origin. As the angle θ sweeps counter-clockwise from the positive x-axis, the point where the angle's ray meets the circle has coordinates (cos θ, sin θ). In other words, sin(θ) is simply the y-coordinate of that point. This is exactly what the diagram above shows: the green dashed line marks the y-distance from the x-axis up to the point on the circle, and that distance is the sine value.

• sin(θ) = Opposite ÷ Hypotenuse (right-triangle definition)
• sin(θ) = y-coordinate of the point on the unit circle at angle θ
• θ = sin⁻¹(x) — the inverse sine (arcsin) function, used to solve for the angle when the ratio is known
• sin(θ) is always between -1 and 1, since it's a ratio on a circle of radius 1

How to Use This Sin Calculator — Step by Step

Pick the mode that matches your problem, enter your numbers, and read the result along with the diagram and worked solution below it.

• sin(θ) from an angle: type an angle and choose degrees, radians, or gradians — the calculator instantly returns sin(θ), cos(θ), tan(θ), the quadrant, and the reference angle, plotted on a unit circle.
• θ from right-triangle sides: enter the opposite side and hypotenuse of a right triangle — the calculator applies sin(θ) = Opposite ÷ Hypotenuse, then uses the inverse sine to solve for θ, and draws the triangle to scale with every side labeled.
• θ from a sine value: enter any decimal between -1 and 1 to solve equations like sin(θ) = 0.5 for θ, returning the principal angle plus the second solution between 0° and 360°.

Worked Example — Solving sin(30°)

Suppose you need sin(30°). Switch to the first mode, enter 30, choose degrees, and the calculator converts 30° to π/6 radians, locates the point on the unit circle, and reads off the y-coordinate: sin(30°) = 0.5 exactly. The diagram shows the radius line reaching a point exactly halfway up the circle's height, matching the well-known result sin(30°) = 1/2.

Now suppose you're given a right triangle with an opposite side of 3 and a hypotenuse of 5 — a classic 3-4-5 triangle. Switching to the second mode and entering those values gives sin(θ) = 3 ÷ 5 = 0.6, and applying the inverse sine, θ = sin⁻¹(0.6) ≈ 36.87°. The calculator also finds the missing adjacent side using the Pythagorean theorem: √(5² − 3²) = 4, completing the full 3-4-5 right triangle.

Common Sine Values Table

These standard angles come up constantly in trigonometry, and it helps to recognize them at a glance rather than recalculating them each time:

• sin(0°) = 0
• sin(30°) = 1/2
• sin(45°) = √2/2
• sin(60°) = √3/2
• sin(90°) = 1
• sin(180°) = 0
• sin(270°) = -1
• sin(360°) = 0

Understanding the Sign of sin(θ) by Quadrant

Because the unit circle wraps all the way around, sin(θ) changes sign depending on which quadrant the angle lands in. This calculator automatically detects and displays the quadrant for any angle you enter:

• Quadrant 1 (0°-90°): sin(θ) is positive — the point is above the x-axis, to the right of the y-axis
• Quadrant 2 (90°-180°): sin(θ) is still positive — the point is above the x-axis, but to the left
• Quadrant 3 (180°-270°): sin(θ) is negative — the point drops below the x-axis
• Quadrant 4 (270°-360°): sin(θ) is negative — the point is below the x-axis, back on the right side

Sine, Cosine, and Tangent — How They Relate

Sine never works in isolation — it's closely tied to cosine and tangent, and this calculator shows all three whenever you solve for an angle. Cosine is the x-coordinate on the same unit circle diagram (the orange dashed segment), and tangent is simply the ratio tan(θ) = sin(θ) ÷ cos(θ). A useful identity worth remembering is sin²(θ) + cos²(θ) = 1, which follows directly from the Pythagorean theorem applied to the unit circle's radius of 1.

Degrees vs Radians vs Gradians

Angles can be measured in more than one unit, and this calculator supports the three most common ones. Degrees split a full circle into 360 equal parts and are the most familiar unit in everyday geometry. Radians measure angle by arc length relative to the radius, making a full circle equal to 2π radians — this is the unit calculus and most programming languages use internally. Gradians (or gons) split a full circle into 400 parts and appear mostly in surveying and some European engineering contexts. Whichever unit your problem uses, simply select it from the dropdown and the calculator handles the conversion automatically.

Where Sine Calculations Are Used in Real Life

Sine isn't just a classroom exercise — it shows up anywhere angles and distances interact:

• Physics: resolving forces, projectile motion, and wave equations (sound, light, and AC electrical signals are all modeled with sine curves).
• Engineering and construction: calculating roof pitches, ramp angles, and structural load components.
• Navigation and surveying: triangulating positions and distances using known angles.
• Computer graphics and game development: rotating objects and animating smooth, wave-like motion.
• Astronomy: calculating the position of celestial objects using spherical trigonometry, which builds directly on the sine rule.

Tips for Getting Accurate Results

Always double-check which angle unit your problem is stated in before entering a value — a common mistake is entering a radian value while the unit dropdown is still set to degrees, which will give a wildly different (and wrong) answer. When solving for an angle from a sine value or triangle sides, remember that inverse sine only returns one principal solution between -90° and 90°; if your problem expects an angle between 0° and 360°, check the second solution shown in the results, since sine repeats and multiple angles can share the same sine value.