Trigonometry Calculator

Forward trig functions: sine, cosine, and tangent

Forward trigonometry starts with an angle and returns a ratio. The sine calculator workflow finds sin(theta), the cosine calculator workflow finds cos(theta), and the tangent calculator workflow finds tan(theta). On the unit circle, sine is the y-coordinate, cosine is the x-coordinate, and tangent is sine divided by cosine.

For right triangles, the same functions describe side ratios. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. That is the SOHCAHTOA memory aid: sin = O/H, cos = A/H, tan = O/A.

The master calculator preserves function-specific intent with anchored panels, exact-value guidance where available, degree/radian switching, quadrant signs, and undefined-value warnings. That keeps searches such as sine calculator, cosine calculator, and tangent calculator useful without maintaining separate competing pages.

Reciprocal trig functions: secant, cosecant, and cotangent

The reciprocal trig functions use the same angle but invert a primary trig value. Cosecant is 1 divided by sine, secant is 1 divided by cosine, and cotangent is 1 divided by tangent. These workflows are useful when a formula is written in reciprocal trig form or when a textbook asks directly for csc, sec, or cot.

Undefined values matter. Secant is undefined wherever cosine is zero, cosecant is undefined wherever sine is zero, and cotangent is undefined wherever tangent has a zero denominator or sine is zero in the reciprocal setup. The calculator surfaces these cases instead of returning a misleading zero.

Inverse trig: arcsin, arccos, arctan, and arccot

Inverse trigonometry reverses the direction of the problem. Instead of starting with an angle, you start with a ratio and solve for an angle. Arcsin answers which angle has a given sine value, arccos answers which angle has a given cosine value, and arctan answers which angle has a given tangent ratio.

Principal values are important for inverse trig calculators. A ratio can often match more than one angle over a full rotation, but inverse functions return a conventional principal angle unless the context asks for a full angle family. The arctan workflow also supports slope-style inputs such as rise/run and percent grade, which are practical forms of tangent.

Arccot is included for formulas and courses that use inverse cotangent directly. It is closely related to arctan, but the principal-value convention and reciprocal setup can differ by source, so the calculator keeps the interpretation visible.

Degrees, radians, reference angles, and coterminal angles

Degrees and radians measure the same rotation in different units. A full turn is 360 degrees or 2*pi radians, so 180 degrees equals pi radians and 90 degrees equals pi/2 radians. The calculator keeps both displays visible where the workflow benefits from conversion.

A reference angle is the acute angle between the terminal side and the x-axis. It helps explain why angles in different quadrants share related exact trig values while their signs differ. For example, 150 degrees has a 30 degree reference angle, so its sine and cosine are connected to the 30 degree special angle values.

Coterminal angles differ by whole rotations. In degrees, add or subtract 360. In radians, add or subtract 2*pi. Coterminal angles have the same trig-function values because they land on the same terminal side.

Unit circle and exact special-angle values

The unit circle connects trig values to coordinates. At an angle theta, the point on the unit circle is (cos(theta), sin(theta)). That means cosine is the x-coordinate and sine is the y-coordinate. Tangent is the ratio y/x where x is not zero.

Special angles such as 0, 30, 45, 60, 90, 180, 270, and 360 degrees have exact values that appear frequently in algebra, calculus, physics, engineering, and test prep. The unit-circle panel is included so users can look up these exact values, quadrant signs, and coordinate pairs without leaving the canonical trigonometry calculator.

Exact values and decimal values both matter. Exact values such as sqrt(2)/2 are best for symbolic math, while decimal values are easier for measurement and numerical work. The page keeps both ideas connected rather than treating each function page as a separate topic.

Choosing the right trig workflow before calculating

The fastest way to avoid a wrong trig answer is to identify the direction of the problem first. If you know an angle and need a ratio, use a forward trig workflow such as sine, cosine, tangent, or the all-functions panel. If you know a ratio and need an angle, use inverse trig such as arcsin, arccos, arctan, or arccot.

If the problem asks for an exact value, a quadrant sign, or a point on the circle, start with the unit-circle or reference-angle workflow before relying on a decimal. If the angle is outside one turn, use coterminal angles to reduce it to the same terminal side, then use the reference angle to explain the sign.

If the input is a right-triangle side setup, use SOHCAHTOA or the right-triangle calculator. If the input is a signed coordinate direction, a single arctan ratio may lose quadrant information because tangent repeats every 180 degrees. In coordinate work, atan2-style logic is the safer model because it keeps both axis signs.

Trig identities and SOHCAHTOA checks

Trig identities are equations that remain true for valid angle inputs. The Pythagorean identity sin^2(theta) + cos^2(theta) = 1 is the foundation for many simplifications. Related reciprocal identities connect csc, sec, and cot back to sine, cosine, and tangent.

SOHCAHTOA is the right-triangle setup workflow. Use it when you know a side and an angle, or a pair of sides, and need to choose the correct sine, cosine, or tangent relationship. The master page includes this workflow but keeps full triangle solvers separate because law-of-sines, law-of-cosines, and general triangle solving require additional cases.

If you need a full geometric triangle solution, use the right triangle calculator, law of sines calculator, law of cosines calculator, or triangle calculator instead. Those pages handle side solving, angle solving, ambiguous-case logic, perimeter, area, and classification in ways that a general trig-function calculator should not hide.

Common mistakes and limitations

The most common mistake is using the wrong angle unit. A value of 1 means 1 degree in degree mode but 1 radian in radian mode, which is about 57.3 degrees. Always check the selected unit before comparing results.

The second common mistake is confusing inverse functions with reciprocal functions. Arcsin is inverse sine and returns an angle. Cosecant is reciprocal sine and returns a ratio. Arctan is inverse tangent, while cotangent is reciprocal tangent.

The calculator is designed for numeric trigonometry, exact-value lookup, identity checks, and right-triangle setup. It does not replace a symbolic algebra system, a proof assistant for identities, or a full triangle solver for non-right triangles with ambiguous geometric cases.