Trigonometric Function Calculator (sin, cos, tan)

Last updated: 2026-05-19

Trigonometric functions

The trigonometric functions relate an angle to ratios of the sides of a right triangle, and more generally to coordinates on the unit circle. The three primary functions are sine, cosine, and tangent; the three reciprocal functions are cosecant, secant, and cotangent. Together these six functions assign a numeric value to every angle, measured in either degrees or radians.

At certain angles a function has no finite value: tangent and secant are undefined where the cosine is zero (90°, 270°, …), and cosecant and cotangent are undefined where the sine is zero (0°, 180°, …). At those angles the calculator reports the undefined case rather than a misleadingly large number.

SOH-CAH-TOA: reading trig functions from a right triangle

The classic mnemonic SOH-CAH-TOA encodes the three primary trig ratios for an acute angle θ in a right triangle:

Function	Mnemonic	Ratio
sin(θ)	Sine = Opposite / Hypotenuse	opp/hyp
cos(θ)	Cosine = Adjacent / Hypotenuse	adj/hyp
tan(θ)	Tangent = Opposite / Adjacent	opp/adj

The three reciprocal functions follow directly:

• csc(θ) = 1/sin(θ) — cosecant
• sec(θ) = 1/cos(θ) — secant
• cot(θ) = 1/tan(θ) = cos(θ)/sin(θ) — cotangent

The unit circle

The unit circle is a circle of radius 1 centered at the origin. For any angle θ measured counterclockwise from the positive x-axis, the corresponding point on the circle is exactly (cos θ, sin θ). This means:

• cos θ is the x-coordinate of that point.
• sin θ is the y-coordinate.
• tan θ is the slope of the line from the origin to the point (rise/run = sin θ / cos θ).

Because the radius is 1, the Pythagorean theorem gives the fundamental identity directly:

sin²(θ) + cos²(θ) = 1

The unit circle extends trig beyond right triangles to any angle — including negative angles (clockwise rotation) and angles beyond 360° — because the rotation simply wraps around the circle.

Worked example: ramp inclination

A construction ramp rises 3.5 m over a horizontal run of 12 m. The angle it makes with the ground, and the trig values at that angle, follow from the side ratios.

Step 1 — find the angle: tan(θ) = opposite/adjacent = 3.5/12 ≈ 0.29167, so θ = arctan(0.29167) ≈ 16.26°

Step 2 — compute the trig values at 16.26°:

Function	Value	Interpretation
sin(16.26°)	≈ 0.2800	The ramp rises 0.28 m per metre of slope length
cos(16.26°)	≈ 0.9600	The horizontal run is 96% of the slope length
tan(16.26°)	≈ 0.2917	Rise-to-run ratio (the slope gradient)
csc(16.26°)	≈ 3.5714	Slope length per metre of rise
sec(16.26°)	≈ 1.0417	Slope length per metre of run
cot(16.26°)	≈ 3.4286	Run-to-rise ratio

Entering 16.26 in degrees reproduces these results.

Exact values for common angles

θ (deg)	θ (rad)	sin	cos	tan
0°	0	0	1	0
30°	π/6	$\tfrac{1}{2}$	$\sqrt{3}/2$	$1/\sqrt{3}$
45°	π/4	$\sqrt{2}/2$	$\sqrt{2}/2$	1
60°	π/3	$\sqrt{3}/2$	$\tfrac{1}{2}$	$\sqrt{3}$
90°	π/2	1	0	undefined
180°	π	0	−1	0
270°	3π/2	−1	0	undefined

Memorizing the 30-45-60 row is usually enough — the rest follow from symmetry and the periodicity of 360° (2π rad).

Periodicity and symmetry

Trig functions repeat with a fixed period:

• sin and cos: period 360° (2π rad) — values repeat every full revolution.
• tan and cot: period 180° (π rad) — they repeat twice as fast.

Key symmetry rules (useful for mental arithmetic):

• sin(−θ) = −sin(θ) — sine is an odd function.
• cos(−θ) = cos(θ) — cosine is an even function.
• sin(180° − θ) = sin(θ) — sine is symmetric about 90°.
• cos(180° − θ) = −cos(θ) — cosine changes sign across 90°.

Applications

Trigonometric values appear wherever an angle has to be turned into a length, a ratio, or a coordinate: surveying and ramp gradients, resolving forces and velocities into components, oscillation and wave models, and the rotation matrices used in graphics and robotics. A trig calculator is most often used to verify a hand computation, sanity-check a formula, or examine how the values change across a range of angles before they are wired into a larger calculation. For inverse problems — recovering the angle from a known ratio — see the Inverse Trigonometric Functions Calculator (arcsin, arccos, arctan).

Frequently Asked Questions (FAQ)

What's the difference between degrees and radians?

Degrees divide a full circle into 360 equal parts — intuitive for everyday geometry. Radians measure angle by arc length: one radian is the angle that subtends an arc equal to the radius, so a full circle is 2π ≈ 6.283 rad. To convert: multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees. Most programming languages (JavaScript, Python, C) use radians for trig functions internally.

Why is tan(90°) undefined?

Tangent equals sin(θ)/cos(θ). At exactly 90°, cos(θ) = 0, so the division produces no finite result — tangent grows without bound as θ approaches 90° from below and from negative infinity as θ approaches from above. This is a vertical asymptote. The same applies at every odd multiple of 90° (270°, 450°, …). Secant = 1/cos θ is also undefined at those angles.

How do I read a unit circle?

The unit circle is a circle of radius 1 centered at the origin. For any angle θ measured counterclockwise from the positive x-axis, the point on the circle is (cos θ, sin θ). Reading the coordinates directly gives cosine (the x-coordinate) and sine (the y-coordinate). Tangent is the slope of the line from the origin to that point: sin θ / cos θ.

The identity sin²θ + cos²θ = 1 follows directly — it's the Pythagorean theorem applied to the unit-circle right triangle.

What are csc, sec, and cot?

They are the three reciprocal trig functions. Cosecant (csc) = 1/sin, secant (sec) = 1/cos, and cotangent (cot) = 1/tan = cos/sin. They appear in integrals, certain physics formulas, and older textbooks. Memory aid: csc is the reciprocal of sin (not cos), and sec is the reciprocal of cos (not sin) — the "co-" prefix denotes the co-function of the complementary angle, not a reciprocal relationship with the same-named function.