Enter any angle in degrees or radians and instantly get all six trigonometric functions — sine, cosine, tangent, cosecant, secant, and cotangent — with the radian conversion shown step by step.
The tool is ideal for homework, engineering checks, geometry problems, and quick conversions. Because every computation happens instantly in your browser, you can try dozens of angles in seconds — compare sin 30°, sin 45°, and sin 60°, and the pattern of the values becomes visible immediately.
Trigonometric functions relate the angles of a triangle to the ratios of its sides. They began as tools for measuring right triangles, but they now appear everywhere: waves, sound, electricity, satellite orbits, computer graphics, and even the compression algorithms behind digital images all run on trigonometry.
Everything starts with the right triangle. Pick one of its acute angles, call it θ, and name the three sides relative to it: the opposite side (across from θ), the adjacent side (next to θ, not the hypotenuse), and the hypotenuse (the longest side, opposite the right angle). The three primary functions are ratios of these sides:
There are also three reciprocal functions: cosecant (csc θ = 1 ÷ sin θ), secant (sec θ = 1 ÷ cos θ), and cotangent (cot θ = 1 ÷ tan θ). They carry no new information — each is simply the flip of one of the primary three — but they make many formulas shorter and are essential in calculus and physics.
The deeper definition comes from the unit circle — a circle of radius 1 centered at the origin. Draw an angle θ from the positive x-axis; the point where its ray meets the circle has coordinates (cos θ, sin θ). This elegant picture explains several big ideas at once: why sine and cosine always stay between −1 and 1, why sin 90° = 1 and cos 90° = 0, why the functions repeat every 360°, and why angles beyond 360° or below 0° still have perfectly good function values. The tangent is the slope of that ray, which is why it shoots to infinity as the ray becomes vertical — the moment we call tan 90° undefined.
Trigonometric functions are periodic: sin(θ + 360°) = sin θ, and the same for cosine, while tangent repeats every 180°. This periodicity is why trig functions model anything that oscillates — a sound wave, a swinging pendulum, alternating current — with a single elegant curve. They also obey the famous Pythagorean identity, sin²θ + cos²θ = 1, which ties the two together for every angle and is the workhorse identity behind thousands of proofs and computations.
The mnemonic SOH-CAH-TOA encodes all three primary definitions: Sine = Opposite over Hypotenuse, Cosine = Adjacent over Hypotenuse, Tangent = Opposite over Adjacent. Memorize this one phrase and you can reconstruct every right-triangle trig formula on demand.
Core formulas
Angles are measured in degrees (360 in a full circle) or radians (2π in a full circle). The conversion is radians = degrees × π / 180. So 180° = π radians, 90° = π/2 radians, and 30° = π/6 radians (approximately 0.523599). This calculator performs the conversion automatically, but showing the step keeps the computation transparent — and it is the single most common source of wrong answers, so double-checking it matters.
Certain angles have famous exact values worth memorizing:
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 = 0.5 | √3/2 ≈ 0.866025 | √3/3 ≈ 0.577350 |
| 45° | √2/2 ≈ 0.707107 | √2/2 ≈ 0.707107 | 1 |
| 60° | √3/2 ≈ 0.866025 | 1/2 = 0.5 | √3 ≈ 1.732051 |
| 90° | 1 | 0 | Undefined |
Worked example 1 — sin 30°. Convert: 30 × π / 180 = π/6 ≈ 0.523599 radians. From the 30-60-90 triangle, the side opposite 30° is half the hypotenuse, so sin 30° = 1/2 = 0.5 exactly. The calculator returns 0.5.
Worked example 2 — cos 60°. 60 × π / 180 = π/3 ≈ 1.047198 radians. In the same triangle, the side adjacent to 60° is half the hypotenuse, so cos 60° = 1/2 = 0.5. Notice the cofunction symmetry: sin 30° = cos 60°.
Worked example 3 — tan 45°. 45 × π / 180 = π/4 ≈ 0.785398 radians. An isosceles right triangle has equal legs, so opposite = adjacent and tan 45° = 1. Equivalently, tan 45° = sin 45° / cos 45° = 0.707107 / 0.707107 = 1.
Worked example 4 — the reciprocal functions at 30°. Since sin 30° = 0.5, csc 30° = 1 / 0.5 = 2. Since cos 30° ≈ 0.866025, sec 30° = 1 / 0.866025 ≈ 1.154701. Since tan 30° ≈ 0.577350, cot 30° = 1 / 0.577350 ≈ 1.732051.
Worked example 5 — tan 90° is undefined. 90 × π / 180 = π/2 ≈ 1.570796 radians. Here cos 90° = 0, and tan 90° = sin 90° / cos 90° = 1 / 0 — division by zero. The calculator flags this as Undefined rather than printing a giant misleading number.
Example 1 — a ladder against a wall. A 5-meter ladder leans against a house, making a 60° angle with the ground. How high up the wall does it reach? The wall is the side opposite the 60° angle, and the ladder is the hypotenuse, so height = 5 × sin 60°. Using the calculator: sin 60° ≈ 0.866025, and 5 × 0.866025 = 4.330125 meters — about 4.33 m. The base of the ladder sits at 5 × cos 60° = 5 × 0.5 = 2.5 meters from the wall. One trig computation answers both practical questions: whether the ladder reaches the 4-meter window and whether its base is a safe distance out.
Example 2 — surveying a building’s height. A surveyor stands 100 meters from a building and measures the angle of elevation to the top of the roof as 35°. The height above eye level is the side opposite the angle and the 100-meter distance is the adjacent side, so height = 100 × tan 35°. The calculator gives tan 35° ≈ 0.700208, so the height is 100 × 0.700208 = 70.0208 meters — about 70 m. Add the surveyor’s eye height (say 1.7 m) for a total building height of roughly 71.7 meters. This is exactly how heights of trees, towers, and cliffs are measured in the field when no one can climb them.
Beyond these textbook cases, the same six functions size roof pitches in construction (tan of the pitch angle gives rise over run), aim antennas and solar panels, convert between map coordinates, and decode the periodic signals in music and medicine. Any time a quantity goes up and down in a repeating cycle, sine and cosine are the natural language for describing it.
Trigonometric Functions Calculator (sigmacalculator.net): interactive web tool computing the six trigonometric functions — sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), cotangent (cot) — for an angle entered in degrees or radians, with automatic degree–radian conversion (radians = degrees × π/180) and explicit flagging of undefined values (e.g., tan 90°). Reference definitions: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = sin θ/cos θ; exact values include sin 30° = 0.5, cos 60° = 0.5, tan 45° = 1. Canonical page: https://sigmacalculator.net/math-calculators/trigonometric-functions-calculator/
sin 30° = 0.5 exactly. In a 30-60-90 right triangle, the side opposite the 30° angle is exactly half the hypotenuse, so opposite ÷ hypotenuse = 1/2. This is one of the standard exact values worth memorizing, along with sin 45° = √2/2 ≈ 0.707107 and sin 60° = √3/2 ≈ 0.866025.
cos 90° = 0. On the unit circle, the point at 90° is (0, 1), and cosine is the x-coordinate. More intuitively, as the angle approaches 90° in a right triangle, the adjacent side shrinks toward zero while the hypotenuse stays fixed, so adjacent ÷ hypotenuse approaches 0. This zero is also the reason tan 90° is undefined — tangent divides by cosine.
SOHCAHTOA is the classic mnemonic for the three primary trig ratios in a right triangle: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. “Opposite” and “adjacent” are always named relative to the angle θ you are solving for. The three reciprocal functions follow directly: csc = 1/sin, sec = 1/cos, cot = 1/tan.
Use whichever unit your problem gives you — but make sure the calculator matches. Degrees (360 per circle) are standard in geometry, construction, and navigation; radians (2π per circle) are standard in calculus, physics, and programming because they make derivative formulas clean. Convert with radians = degrees × π/180. The most common trig mistake in the world is a degree/radian mode mismatch, so always verify the unit before trusting a result.
Because tan θ = sin θ / cos θ, and cos 90° = 0 — division by zero is undefined. Geometrically, the tangent is the slope of the ray at angle θ, and at 90° the ray points straight up: its slope is infinite, which has no real-number value. Tangent is undefined at every odd multiple of 90° (90°, 270°, 450°, and so on), and the secant shares the same undefined points since sec θ = 1/cos θ.
The unit circle is a circle of radius 1 centered at the origin. For an angle θ measured from the positive x-axis, the point where the angle’s ray meets the circle is exactly (cos θ, sin θ). This single picture explains why sine and cosine range between −1 and 1, why the functions repeat every 360°, how negative and large angles work, and where the exact values for 30°, 45°, and 60° come from.
Memorize the 0°–90° table: sin 0° = 0, sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2, sin 90° = 1; cosine runs the same list in reverse (cos 0° = 1 down to cos 90° = 0); tan 0° = 0, tan 30° = √3/3, tan 45° = 1, tan 60° = √3, tan 90° = undefined. The reciprocal functions follow by flipping: csc 30° = 2, sec 60° = 2, cot 45° = 1.
Inverse trig functions reverse the direction: instead of angle → ratio, they compute ratio → angle. arcsin(0.5) = 30°, arccos(0.5) = 60°, arctan(1) = 45°. They answer questions like “at what angle does the ladder reach 4.33 meters up a 5-meter ladder?” — arcsin(4.33/5) ≈ 60°. Try them on our Inverse Trigonometric Functions Calculator.
Cite this page as: Trigonometric Functions Calculator — Sigma Calculator. https://sigmacalculator.net/math-calculators/trigonometric-functions-calculator/
Last updated: October 11, 2026