Trigonometry Calculator

Getting started is straightforward. Select the function you want to calculate, enter your known values, and hit calculate. The tool handles the math and returns your result instantly.

A few things worth knowing before you dive in:

• Choose your angle mode first. Decide whether you're working in degrees or radians. Most high school problems use degrees; calculus and advanced math often use radians. Mixing them up is a common source of wrong answers.
• Enter what you know. Depending on the function, you might enter an angle to get a ratio, or enter a ratio to get an angle (that's what inverse functions are for).
• Read the output carefully. The calculator returns the numerical result, but for triangle solvers it also shows the full set of solved values so you can see the complete picture.

If you're not sure which function to use, check the sections below. Each one explains what the function does and when to reach for it.

Sine, cosine, and tangent are the three core trigonometric functions. Each one describes a specific ratio between two sides of a right triangle, based on a given angle.

• Sine (sin) of an angle equals the length of the opposite side divided by the hypotenuse.
• Cosine (cos) of an angle equals the length of the adjacent side divided by the hypotenuse.
• Tangent (tan) of an angle equals the length of the opposite side divided by the adjacent side.

To use the calculator, enter an angle value and select the function. For example, sin(30°) = 0.5, cos(60°) = 0.5, and tan(45°) = 1. These are exact values that come up constantly in trig problems.

These functions work for any angle, not just the tidy ones on a unit circle. The calculator returns decimal approximations for angles like 37° or 1.2 radians just as easily. That's where having a tool like this really pays off.

A right triangle has one 90° angle, two other angles, and three sides. If you know at least two of those values (with at least one being a side length), you can find everything else. That's exactly what the right triangle solver does.

Enter the values you already know and the calculator fills in the rest. It solves for missing sides, missing angles, and will even confirm that your inputs form a valid triangle. This is especially useful in geometry, physics, and any kind of construction or design work where measurements matter.

Regular trig functions take an angle and return a ratio. Inverse trig functions do the opposite: you give them a ratio, and they return the angle that produces it.

These are essential whenever you're solving for an unknown angle rather than an unknown side. They show up throughout geometry, physics, and engineering, and they're used constantly in the right triangle solver under the hood.

One thing to keep in mind: inverse trig functions have restricted output ranges by convention. Arcsin and arctan return values between -90° and 90°, while arccos returns values between 0° and 180°. This doesn't mean other solutions don't exist; it just reflects the standard principal value range.

Angles can be measured in degrees or radians. Degrees are more intuitive for most people (a full circle is 360°), while radians are the standard unit in higher math and science (a full circle is 2π radians).

The conversion formulas are simple:

• To convert degrees to radians: multiply by π/180
• To convert radians to degrees: multiply by 180/π

A few common conversions worth memorizing:

The calculator lets you switch between modes at any time. Just make sure your inputs match the selected mode before calculating, otherwise your results will be off.

Trig identities are equations involving trig functions that hold true for all valid angle values. They're used to simplify expressions, solve equations, and prove other mathematical relationships. Knowing the key ones makes a lot of trig work go faster.

Some of the most commonly used identities include the Pythagorean identities (sin²θ + cos²θ = 1 is the big one), the reciprocal identities (cosecant, secant, and cotangent), the quotient identities (tan = sin/cos), and the double-angle formulas used in calculus and physics.

You don't need to memorize all of them at once. Start with the foundational ones and build from there as the problems you work on start demanding more.

Seeing how the math works through concrete examples makes things click a lot faster than reading definitions alone. Here are a few typical problems with step-by-step solutions.

Example 1: Find sin(45°)
sin(45°) = √2/2 ≈ 0.7071. This is a standard special angle value that appears constantly in trig.

Example 2: Solve for a missing side
A right triangle has an angle of 30° and a hypotenuse of 20. Find the opposite side.
opposite = 20 × sin(30°) = 20 × 0.5 = 10.

Example 3: Find a missing angle
The adjacent side is 8 and the hypotenuse is 10. What's the angle?
cos θ = 8/10 = 0.8, so θ = arccos(0.8) ≈ 36.87°.

Example 4: Convert 135° to radians
135 × (π/180) = 3π/4 ≈ 2.356 radians.

These examples cover the core scenarios. Most trig problems you'll encounter are variations on one of these four patterns.

The unit circle is a circle with a radius of 1 centered at the origin on a coordinate plane. It's one of the most useful tools in trigonometry because it defines sine and cosine values for every angle in a clean, visual way.

On the unit circle, for any angle θ, the coordinates of the point on the circle are (cos θ, sin θ). That's it. The x-coordinate is the cosine and the y-coordinate is the sine.

The special angles (0°, 30°, 45°, 60°, 90°, and their equivalents in other quadrants) have exact values that you'll see over and over:

Once you're comfortable with the first quadrant, the other three follow from symmetry. Memorizing these values pays dividends in exams and in any math-heavy field.

Trig isn't just a classroom subject. It shows up in a surprisingly wide range of fields, often in ways you wouldn't immediately recognize.

• Architecture and construction: Calculating roof pitches, ramp angles, and structural loads all involve trig ratios. A roof with a 6/12 pitch has a specific angle that engineers calculate using arctangent.
• Navigation: Pilots and sailors use trigonometry to calculate headings, distances, and course corrections. GPS systems rely on trigonometric calculations to pinpoint locations.
• Physics: Projectile motion, wave behavior, and force vector decomposition all require sine and cosine. Breaking a force into horizontal and vertical components is pure SOH CAH TOA.
• Engineering: Mechanical and electrical engineers use trig constantly, from designing gears and linkages to analyzing AC circuits (where phase angles matter a lot).
• Computer graphics: Rotating objects, calculating lighting angles, and rendering 3D scenes all depend on trig functions running thousands of times per second.
• Music and acoustics: Sound waves are modeled as sine waves. Audio engineers use trigonometry to analyze frequency, phase, and interference patterns.

The more you work with trig, the more you'll notice it hiding in everyday problems. It's one of those areas of math that keeps showing up no matter what field you're in.

Beyond sine, cosine, and tangent, there are three reciprocal functions that come up in more advanced trig work. Each one is simply the inverse ratio of one of the main three.

Cosecant, secant, and cotangent don't appear as often in basic trig, but they're used regularly in calculus (especially in integrals) and in physics. Knowing their definitions and how they relate to the main three functions makes it much easier to work through problems that involve them.

One quick tip: if you ever see csc, sec, or cot and you're not sure what to do, just rewrite them in terms of sin, cos, or tan. That single step simplifies most expressions and makes the path forward a lot clearer.