What is Trigonometry?
Trigonometry is the branch of mathematics that studies the relationship between the angles and side lengths of triangles. It starts with the right triangle, where six ratios — sine, cosine, tangent, cotangent, secant and cosecant — connect an angle to the lengths of the triangle's sides. Those same six functions extend naturally to any angle, which is why trigonometry underpins geometry, physics, engineering, navigation and computer graphics.
How to Use the Trigonometry Calculator
Pick a function (sin, cos, tan, cot, sec or cosec), type an angle, and choose whether it's in degrees or radians — or just type a short expression like sin 30° or tan(45) into the quick-entry box and press Parse. Press Calculate to see both the exact value (as a fraction or radical, when the angle is a standard one) and the decimal value. Press "Show Steps" for a worked derivation.
Standard Trigonometric Values
The angles 0°, 30°, 45°, 60° and 90° are called "standard angles" because their trigonometric ratios reduce to simple fractions and square roots — 30°, 45° and 60° come directly from cutting an equilateral triangle in half or from a right-isosceles triangle. Because the values are exact, teachers and exams use them constantly, and it's worth memorizing the table above rather than reaching for a calculator every time.
Architecture & Construction
Roof pitches, staircases, and structural supports are all calculated using right-triangle trigonometry.
Navigation & Surveying
Bearings and distances between two known points are found with the sine and cosine rules.
Waves & Physics
Sound, light and AC electricity are all modeled with sine and cosine functions.
Graphics & Game Dev
Rotations, camera angles and lighting in 3D graphics rely on trigonometric functions.
SOH-CAH-TOA
SOH-CAH-TOA is the classic memory aid for the three primary ratios in a right triangle, relative to an angle θ:
CAH: cos θ = Adjacent ÷ Hypotenuse
TOA: tan θ = Opposite ÷ Adjacent
The remaining three functions are simply the reciprocals: cosecant flips sine, secant flips cosine, and cotangent flips tangent.
Trigonometric Identities
These identities hold true for every angle where the functions involved are defined, and they're the algebra behind the "Show Steps" explanations above.
Right Triangle Formulas
Alongside SOH-CAH-TOA, every right triangle obeys the Pythagorean theorem, which relates the two legs to the hypotenuse:
Use the Right Triangle Calculator above: enter the Opposite and Adjacent sides, and it returns the Hypotenuse, all six ratios, and the angle θ = tan⁻¹(Opposite ÷ Adjacent).
Sine Rule
The sine rule works for any triangle, not just right triangles. It states that the ratio of a side to the sine of its opposite angle is the same for all three sides:
It's most useful when you know two angles and one side (AAS/ASA), or two sides and a non-included angle (SSA). Use the Sine Rule Calculator above to solve for a missing side or angle.
Cosine Rule
The cosine rule generalizes the Pythagorean theorem to any triangle. It has three equivalent forms, one for each side:
b² = a² + c² − 2ac cos B
c² = a² + b² − 2ab cos C
It's the right tool when you know two sides and the included angle (SAS) — to find the third side — or all three sides (SSS) — to find any angle. Both cases are available in the Cosine Rule Calculator above.
Formula Reference
A compact summary for revision:
Reciprocals: cosec=1/sin, sec=1/cos, cot=1/tan
Pythagorean: sin²θ+cos²θ=1, sec²θ−tan²θ=1, cosec²θ−cot²θ=1
Sine Rule: a/sinA = b/sinB = c/sinC
Cosine Rule: a² = b²+c²−2bc·cosA
Frequently Asked Questions
A trigonometry calculator finds the sine, cosine, tangent, cotangent, secant, or cosecant of an angle. CalcHub Pro's version gives an exact answer (like 1/2 or √3/3) for standard angles, plus a decimal value, so you can check homework or verify a step in a larger problem.
Use tan θ = sin θ / cos θ. At 30°, sin 30° = 1/2 and cos 30° = √3/2, so tan 30° = (1/2) ÷ (√3/2) = 1/√3, which rationalizes to √3/3 ≈ 0.5774.
sin 30° = 1/2, exactly 0.5. It is one of the five standard angles every trigonometry student memorizes.
cos 60° = 1/2, exactly 0.5. Note that cos 60° equals sin 30° — the two angles are complementary (they add up to 90°).
tan 45° = 1. At 45°, sin and cos are equal (both √2/2), so their ratio is exactly 1.
0°, 30°, 45°, 60° and 90° are the standard angles. Their sine, cosine and tangent values come out as simple fractions or square roots because they are derived from equilateral and right-isosceles triangles, so no calculator or approximation is needed.
Degrees split a full circle into 360 equal parts. Radians measure an angle by the arc length it cuts on a unit circle, where a full circle equals 2π radians. To convert, radians = degrees × π / 180.
tan θ = sin θ / cos θ, and cos 90° = 0. Dividing by zero has no defined result, so tan 90° (and sec 90°) is undefined rather than infinite.
SOH-CAH-TOA is a memory aid for right-triangle ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
Sine, cosine and tangent, plus their reciprocals: cosecant (1/sin), secant (1/cos), and cotangent (1/tan).