Trigonometry Calculator
Calculate sine, cosine, tangent and their inverses. Choose between degrees and radians.
Quick Examples
Trigonometric Functions
Trigonometric functions relate angles to sides of a right triangle.
Basic Definitions
cos(θ) = adjacent/hypotenuse
tan(θ) = opposite/adjacent = sin(θ)/cos(θ)
Common Values
| θ (degrees) | sin(θ) | cos(θ) | tan(θ) |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
Pythagorean Identity
All Six Functions and Where They Are Undefined
| Function | Definition | Undefined where |
|---|---|---|
| sin(θ) | opposite ÷ hypotenuse | Never |
| cos(θ) | adjacent ÷ hypotenuse | Never |
| tan(θ) | sin ÷ cos | cos = 0: 90°, 270° |
| csc(θ) | 1 ÷ sin | sin = 0: 0°, 180° |
| sec(θ) | 1 ÷ cos | cos = 0: 90°, 270° |
| cot(θ) | cos ÷ sin | sin = 0: 0°, 180° |
Note that cot(90°) = 0, not undefined: cot is cos ÷ sin = 0 ÷ 1. Each function fails only where its own denominator is zero.
Exact Values for Special Angles
| θ | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° (π/6) | 1/2 | √3/2 | 1/√3 |
| 45° (π/4) | √2/2 | √2/2 | 1 |
| 60° (π/3) | √3/2 | 1/2 | √3 |
| 90° (π/2) | 1 | 0 | undefined |
Memory trick: the sine column reads √0/2, √1/2, √2/2, √3/2, √4/2 — which simplifies to 0, 1/2, √2/2, √3/2, 1. The cosine column is the same list read upward. A calculator shows 0.707107 where an exam wants √2/2; they are the same number, one exact and one rounded.
Quadrant Signs: The ASTC Rule
Beyond 90°, signs follow the quadrant the angle lands in. The mnemonic ASTC (All Students Take Calculus) tells you which functions are positive:
| Quadrant | Angles | Positive functions | Example |
|---|---|---|---|
| I | 0°–90° | All six | sin 30° = 0.5 |
| II | 90°–180° | sin, csc | cos 120° = −0.5 |
| III | 180°–270° | tan, cot | tan 225° = 1 |
| IV | 270°–360° | cos, sec | cos 300° = 0.5 |
The Unit Circle View
An equivalent way to see everything: place the angle's vertex at the origin and measure along a circle of radius 1. The point where the angle's ray meets the circle is (cos θ, sin θ) — so cosine is simply the x-coordinate and sine the y-coordinate. That single fact explains the quadrant signs, why sin²θ + cos²θ = 1 (Pythagoras on the coordinates), and why angles beyond 360° repeat the same values.
Degrees vs Radians
degrees = radians × 180 / π
Use degrees for geometry and construction; use radians for calculus and physics — the derivative of sin(x) is cos(x) only when x is in radians. A full turn is 2π radians (≈ 6.2832), and 1 radian ≈ 57.2958°.
Identities You Will Actually Use
tan θ = sin θ / cos θ
sin(90° − θ) = cos θ
The first is the check to reach for when verifying an answer: square the sine and cosine and they must add to 1. The third explains why the sine and cosine columns of the exact-values table are mirror images.
Worked Examples
- Tree height: stand 20 m from a tree and measure 35° to the top. Height above eye level = 20 × tan 35° ≈ 14.00 m.
- Ladder: a 6 m ladder leaning at 75° reaches 6 × sin 75° ≈ 5.80 m up the wall, with its foot 6 × cos 75° ≈ 1.55 m out.
- Ramp angle: a ramp rising 1 m over a 12 m run sits at arctan(1/12) ≈ 4.76°.
Inverse Trig Returns One Angle
arcsin(0.5) returns 30°, but 150° has the same sine — inverse functions return the principal value only. Also note that sin⁻¹ means inverse sine (arcsin), not 1/sin(x); the reciprocal of sine is csc(x).
Common Mistakes
- Degrees vs radians: sin(30) is 0.5 in degrees but −0.988 in radians. Check sin(30) = 0.5 before trusting anything else.
- Mixing up opposite and adjacent: both are defined relative to the angle you chose, not fixed sides.
- Using SOH CAH TOA on a non-right triangle: for oblique triangles use the Law of Sines or Law of Cosines.
- Reading 0.707107 as different from √2/2: same value, one rounded.
- Forgetting the second angle from an inverse: arcsin(0.5) = 30°, but 150° is the other valid angle.
Worked Example: Solving a Right Triangle
Given a hypotenuse of 10 and an angle of 30°:
- Opposite side: 10 × sin(30°) = 10 × 0.5 = 5.
- Adjacent side: 10 × cos(30°) = 10 × (√3/2) ≈ 8.66.
- Other angle: 90° − 30° = 60°.
Even–Odd and Periodic Properties
cos(−θ) = cos(θ) (even function)
sin(θ + 2π) = sin(θ), cos(θ + 2π) = cos(θ), tan(θ + π) = tan(θ)
The evenness of cosine explains the mirror-image columns in the exact-values table, and periodicity means trig functions repeat forever — which is why they model waves and oscillation.
Why Radians Are the Natural Unit
On a circle of radius r, the length of an arc spanning θ radians is simply r·θ — with degrees you would need an extra π/180 factor. That simplicity is why calculus, physics, and this calculator's radian mode all prefer radians: the geometry is cleaner.
Where Trigonometry Shows Up
- Navigation: bearings, GPS triangulation, and course corrections.
- Waves and sound: anything sinusoidal — alternating current, signal processing, music.
- Engineering: force decomposition, roof pitches, gear design, optics.
- Computer graphics: rotation matrices, 3D projections, animation.
Sum, Difference, and Double-Angle Identities
cos(A ± B) = cos A·cos B ∓ sin A·sin B
sin(2θ) = 2·sin θ·cos θ
cos(2θ) = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1
Example: sin(75°) = sin(45° + 30°) = sin45°·cos30° + cos45°·sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = √2(√3 + 1)/4 ≈ 0.9659 — the exact form of what the calculator shows as a decimal.
Deriving One Identity from Another
You do not need to memorize every identity separately. The double-angle formulas fall out of the sum formulas (set A = B = θ), the co-function identity sin(90° − θ) = cos θ follows from the difference formula, and tan(A + B) = sin(A + B)/cos(A + B) expands directly. Learn the sum formulas cold; the rest are consequences.
Worked Example: Verifying an Identity
Show that (1 − cos²θ)/sin θ = sin θ for sin θ ≠ 0:
- Start from the Pythagorean identity: 1 − cos²θ = sin²θ.
- Substitute: sin²θ/sin θ.
- Simplify: sin θ. Both sides match.
The standard verification strategy: work from the more complicated side toward the simpler one, using the Pythagorean identity first whenever you see 1 − cos²θ or 1 − sin²θ.
Reference Angles
Any angle's trig values reduce to an acute reference angle plus a quadrant sign. For 150°: the reference angle is 180° − 150° = 30°, in quadrant II where sine is positive, so sin(150°) = sin(30°) = 0.5. For 210°: reference 30°, quadrant III where both are negative, so cos(210°) = −cos(30°) = −√3/2.