Trigonometry Calculator

How to Use This Calculator

Pick a mode from the tabs, enter your values, and click Calculate.

1. Trig Functions – Enter any angle to get sine, cosine, tangent, and their reciprocals. Works with degrees, radians, or gradians. Use the quick-select buttons for common angles.
2. Inverse Trig – Enter a trig value to find the angle. Runs arcsin, arccos, arctan, and the reciprocal inverses. Checks domain restrictions automatically.
3. Right Triangle – Enter any two measurements (at least one side) and the solver finds everything else using the Pythagorean theorem and trig ratios.
4. Oblique Triangle – For triangles without a right angle. Pick your configuration (SSS, SAS, ASA, AAS, or SSA) and the calculator applies Law of Sines or Law of Cosines. SSA cases check for the ambiguous case.
5. Unit Circle – Visual reference showing coordinates at standard angles. Toggle degrees, radians, or exact values on and off.

The Six Trigonometric Functions

Six functions sit at the core of trig, connecting angles to side ratios in right triangles.

On the unit circle, sine (sin) tracks the y-coordinate as you rotate around. For right triangles, that's opposite over hypotenuse. Sine accepts any angle but only outputs values from -1 to 1, swinging between those limits as the angle grows.

Cosine (cos) works like sine, but tracks the x-coordinate instead—adjacent over hypotenuse in triangle terms. It shares the same [-1, 1] range. Cosine runs 90° ahead of sine: cos(θ) = sin(θ + 90°).

What makes tangent (tan) different? It's the ratio of sine to cosine (opposite over adjacent), so it can spit out any real number. But when cosine hits zero—at 90°, 270°, and so on—tangent blows up. Those are its vertical asymptotes.

Flip those three and you get the reciprocals: cosecant (csc) = 1/sin, secant (sec) = 1/cos, cotangent (cot) = 1/tan. They blow up wherever the original function hits zero.

Understanding Sine, Cosine, and Tangent

SOH-CAH-TOA is the classic mnemonic for right triangle ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. Only works for acute angles in right triangles.

First, identify which side is which. Opposite sits directly across from your angle, never touching it. Adjacent shares a vertex with your angle (along with the hypotenuse). The hypotenuse? Always the longest side, always across from the 90° corner.

The unit circle stretches these definitions beyond acute angles. Picture a radius-1 circle at the origin. Any angle θ—measured counterclockwise from the positive x-axis—lands you at a point on that circle. Your x-coordinate there is cos(θ). Your y-coordinate is sin(θ).

That's why sine and cosine stay trapped between -1 and 1: no point on a radius-1 circle can stray further. It also explains why trig functions repeat—one full lap around the circle brings you back to where you started.

The Unit Circle

The unit circle is the single most useful diagram in trig. Take a circle with radius 1, center it at the origin, and every point on it sits at coordinates (cos θ, sin θ).

It ties triangle trig to circular motion. Because the radius is 1, arc length equals angle in radians—that's why calculus and physics prefer radians over degrees.

Memorizing the unit circle means knowing coordinates at key angles. At 0° you're at (1, 0). At 90° (π/2), you're at (0, 1). At 180° (π), you're at (-1, 0). At 270° (3π/2), you're at (0, -1). Those four points mark where the circle crosses the axes.

Between them sit the special angles from 30-60-90 and 45-45-90 triangles. At 30° (π/6): (√3/2, 1/2). At 45° (π/4): (√2/2, √2/2). At 60° (π/3): (1/2, √3/2). Other quadrants keep the same numbers but flip signs based on which coordinates go negative.

Special Angles and Exact Values

Certain angles produce trig values you can write exactly with fractions and square roots—no decimals needed. These special angles (0°, 30°, 45°, 60°, 90°, and their counterparts in other quadrants) show up all the time in coursework.

Where do the values come from? Two special right triangles. A 45-45-90 triangle has equal legs and a hypotenuse √2 times longer. With legs of length 1, the hypotenuse is √2, so sin(45°) = cos(45°) = 1/√2 = √2/2.

Bisect an equilateral triangle and you get a 30-60-90. Short leg 1, hypotenuse 2, long leg √3. That gives sin(30°) = 1/2, cos(30°) = √3/2, sin(60°) = √3/2, cos(60°) = 1/2.

Memorization trick: for sine at 0°, 30°, 45°, 60°, 90°, the pattern is √0/2, √1/2, √2/2, √3/2, √4/2—which simplifies to 0, 1/2, √2/2, √3/2, 1. Cosine runs the same sequence backward.

Reference Angles and Quadrants

To find trig values in any quadrant, first find the reference angle—the acute angle between your terminal side and the x-axis. Always between 0° and 90°.

Quadrant I: reference angle = the angle itself. Quadrant II: subtract from 180°. Quadrant III: subtract 180°. Quadrant IV: subtract from 360°.

Once you have the reference angle, you know the magnitude. But which sign? The mnemonic "All Students Take Calculus" tells you which functions stay positive in each quadrant: All in I, Sine in II, Tangent in III, Cosine in IV. The reciprocals follow their partners—cosecant with sine, secant with cosine, cotangent with tangent.

Why? Sine = y-coordinate, positive where y > 0 (Quadrants I and II). Cosine = x-coordinate, positive where x > 0 (I and IV). Tangent = sine/cosine, positive when both have the same sign (I and III).

Inverse Trigonometric Functions

Inverse trig functions run the calculation backward. sin(30°) = 0.5 asks "what's the sine of 30°?" while arcsin(0.5) asks "what angle gives a sine of 0.5?" Answer: 30°. But there's a catch.

Multiple angles share the same trig value—both 30° and 150° have sine = 0.5. Mathematicians restrict the output to one range (the principal value) so each input returns exactly one answer. For arcsin and arctan, that's -90° to 90°. For arccos, it's 0° to 180°.

Inputs are restricted too. Sine and cosine only produce values from -1 to 1, so arcsin and arccos only accept that range. arcsin(2) throws an error—no angle has a sine of 2. arcsec and arccsc need inputs with absolute value ≥ 1.

Calculators return only the principal value. If sin(θ) = 0.5, your calculator gives θ = 30°. But 150° also works (sine is positive in Quadrant II). Add 360° multiples for infinitely many solutions.

Degrees vs Radians

Degrees split a circle into 360 parts. Radians measure angles by arc length—1 radian covers an arc equal to the radius. Full circle = 2π radians ≈ 6.283.

Converting between them is simple. Degrees to radians: multiply by π/180. Radians to degrees: multiply by 180/π. Memorize these: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 360° = 2π.

Radians matter in calculus because they simplify derivatives and integrals. The derivative of sin(x) equals cos(x) only when x is in radians—using degrees would clutter the formulas with conversion factors.

For everyday work, degrees often feel more natural. Engineers describe a 30° slope; physicists call it π/6 radians. Most calculators and programming languages default to radians, so check your mode before computing.

Right Triangle Trigonometry

Solving a right triangle means finding all sides and angles when you only know a few. That fixed 90° angle helps—the other two must add to 90°. With the Pythagorean theorem and basic trig ratios, any two measurements (including at least one side) pin down everything else.

a² + b² = c² relates the three sides. Know any two? You can find the third. Trig ratios link sides to angles: sin(A) = a/c, cos(A) = b/c, tan(A) = a/b. Flip these around with inverse functions to get angles from side ratios.

Got one side and one acute angle? Find the other angle first (subtract from 90°). Then trig ratios give you the unknown sides. Example: if you know angle A and hypotenuse c, then a = c·sin(A) and b = c·cos(A).

Right triangle problems show up everywhere. Building height from shadow? tan(elevation) = height/shadow. Ramp slope? sin(θ) = rise/length. Navigation? Often just right triangle math with distances and bearings.

Law of Sines

In any triangle, a/sin(A) = b/sin(B) = c/sin(C)—each side divided by the sine of its opposite angle gives the same number. That constant equals 2R, where R is the circumradius (the radius of the circle through all three vertices).

When do you use it? Whenever you have angle-side pairs: AAS, ASA, or SSA. For AAS and ASA, find the third angle first (they sum to 180°), then apply the law to get missing sides.

It works through proportions. Know A, B, and side a? Then b = a·sin(B)/sin(A). Same rearrangement gets you side c. One formula, multiple unknowns.

Limitation: Law of Sines can't touch SSS or SAS directly—too many unknowns in the proportion. Those need Law of Cosines. And SSA cases require extra care because of the ambiguous case (multiple triangles might fit the data).

Law of Cosines

c² = a² + b² - 2ab·cos(C)—the Law of Cosines extends the Pythagorean theorem to all triangles. When C = 90°, cos(C) = 0 and you're left with c² = a² + b². The extra term handles the non-right cases.

Use it for SSS (three sides) or SAS (two sides plus the included angle). For SAS, plug in and compute the third side. For SSS, rearrange: cos(C) = (a² + b² - c²)/(2ab), then take arccos to get each angle.

Three symmetric forms exist—one per side: a² = b² + c² - 2bc·cos(A), b² = a² + c² - 2ac·cos(B), c² = a² + b² - 2ab·cos(C). Pick whichever has the unknown you're after.

For SSS, a good strategy: find the largest angle first (opposite the longest side). Gets around issues with obtuse angles and rounding errors that can trip you up if you switch to Law of Sines too early.

The Ambiguous Case (SSA)

Two sides plus a non-included angle (SSA) is where things get messy. You might get zero, one, or two valid triangles from the same inputs. That's the "ambiguous case."

Picture it geometrically. You have sides a and b, with angle A opposite side a. Swing side a like a compass from point B. Depending on how long a is compared to the height h = b·sin(A), it might miss side b entirely, touch it once, or cut through it twice.

Here are the rules: a < h means no triangle (side's too short). a = h means exactly one right triangle. h < a < b means two triangles. a ≥ b means one triangle (side's long enough to skip the ambiguity).

If two triangles exist, you'll get two angles B from arcsin: the principal value and 180° minus that. Each gives a different triangle with different C and c values. Both are mathematically valid—context decides which applies to your problem.

Trigonometric Identities

Trig identities are equations that work for any angle—they're how you simplify expressions and solve equations. The core ones come straight from definitions and the Pythagorean theorem.

Start with sin²θ + cos²θ = 1—that's the Pythagorean theorem rewritten for the unit circle (x² + y² = 1 becomes cos²θ + sin²θ = 1). Divide through by cos²θ to get 1 + tan²θ = sec²θ. Divide by sin²θ for 1 + cot²θ = csc²θ. Three Pythagorean identities from one idea.

Reciprocal identities are just definitions: csc(θ) = 1/sin(θ), sec(θ) = 1/cos(θ), cot(θ) = 1/tan(θ). Quotient identities express tan and cot as ratios: tan(θ) = sin(θ)/cos(θ), cot(θ) = cos(θ)/sin(θ).

What happens when you negate the angle? Cosine doesn't change: cos(-θ) = cos(θ)—it's even. Sine and tangent flip sign: sin(-θ) = -sin(θ), tan(-θ) = -tan(θ)—they're odd. For complementary angles, functions swap: sin(90° - θ) = cos(θ), tan(90° - θ) = cot(θ), sec(90° - θ) = csc(θ).

Sum, Difference, and Double Angle Formulas

Need sin(A + B)? You can express it using sin and cos of A and B separately: sin(A + B) = sin(A)cos(B) + cos(A)sin(B). For subtraction: sin(A - B) = sin(A)cos(B) - cos(A)sin(B). Cosine works similarly: cos(A + B) = cos(A)cos(B) - sin(A)sin(B) and cos(A - B) = cos(A)cos(B) + sin(A)sin(B).

You'll use these to find exact values for angles like 15° (= 45° - 30°) or 75° (= 45° + 30°). Calculus uses them to integrate trig products; physics uses them for wave interference.

Set A = B and you get double angle formulas: sin(2θ) = 2sin(θ)cos(θ). For cosine, you have three equivalent forms: cos(2θ) = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ. Pick whichever fits your problem. tan(2θ) = 2tan(θ)/(1 - tan²θ).

Half angle formulas go the other direction: sin(θ/2) = ±√((1 - cos θ)/2), cos(θ/2) = ±√((1 + cos θ)/2). The ± depends on which quadrant θ/2 lands in. You'll see them in integration problems and signal processing.

Applications of Trigonometry

Trig shows up anywhere you find angles, waves, or triangles:

• Surveying and Navigation – Measure angles to distant points, then Law of Sines or Cosines gives you distances. GPS triangulates from satellites the same way.
• Physics is full of trig. Projectile motion? Horizontal = cosine, vertical = sine. Sound waves, light waves, AC current—all sine functions. Any time you break a force into components, you're using trig ratios.
• Engineering – AC circuits run on sine waves for voltage and current. Structural engineers resolve forces in beams. Signal processing uses Fourier analysis to decompose complex signals into sine components—everything from audio compression to MRI imaging depends on this.
• Computer Graphics – Rotations are built on sine and cosine. Game engines calculate camera angles, projectiles, collisions. Animation software moves objects along curved paths.

Common Trigonometry Mistakes

Degree/Radian Mix-ups

Most calculators and programming languages default to radians. Typing sin(90) returns about 0.894, not 1, because that's 90 radians. Check your mode before calculating.

Inverse Notation Confusion

sin⁻¹(x) means arcsin(x), not 1/sin(x). The reciprocal of sine is cosecant. Same pattern: cos⁻¹ is arccos (not secant), tan⁻¹ is arctan (not cotangent).

Domain Errors

Inverse functions have restricted inputs. arcsin(1.5) doesn't exist—no angle has a sine greater than 1. arctan accepts any number, but arcsec and arccsc need inputs with absolute value of at least 1.

Sign Errors by Quadrant

Reference angles give magnitude, but you still need the correct sign for each quadrant. Cosine is negative in Quadrants II and III. Sine is negative in III and IV. Use "All Students Take Calculus" to remember which functions stay positive where.

Pythagorean Theorem on Non-Right Triangles

a² + b² = c² only works when angle C is 90°. For other triangles, use the Law of Cosines: c² = a² + b² − 2ab·cos(C).