sin(45°)
0.707107
cos
0.7071
tan
1.0000
0.707107
0.707107
1
1.414214
1.414214
1
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 0.5 | 0.8660 | 0.5774 |
| 45° | 0.7071 | 0.7071 | 1 |
| 60° | 0.8660 | 0.5 | 1.7321 |
| 90° | 1 | 0 | undef |
sin(θ) = opp/hyp, cos(θ) = adj/hyp, tan(θ) = opp/adjThe three primary trigonometric ratios relate the sides of a right triangle to its angles. These are the building blocks for all trigonometric calculations.
Where:
θ= The angle being evaluatedopp= Side opposite to the angleadj= Side adjacent to the anglehyp= Hypotenuse (longest side, opposite 90°)radians = degrees × π / 180Converts an angle from degree measure to radian measure. One full revolution equals 360 degrees or 2π radians.
Where:
degrees= Angle in degrees (0° to 360°)π= Pi, approximately 3.14159radians= Angle in radianssin²(θ) + cos²(θ) = 1The fundamental trigonometric identity. For any angle, the square of sine plus the square of cosine always equals 1. Derived from the unit circle and the Pythagorean theorem.
Where:
θ= Any angle in degrees or radianssin²= Sine value squaredcos²= Cosine value squaredInputs
Result
sin(30°) = 0.5 is one of the exact values from the unit circle. In a 30-60-90 triangle, the side opposite 30° is exactly half the hypotenuse.
Inputs
Result
cos(60°) = 0.5 is a complementary identity: cos(60°) = sin(30°). In a 30-60-90 triangle, the adjacent side to 60° is half the hypotenuse.
Inputs
Result
tan(45°) = 1 because sin(45°) = cos(45°). In a 45-45-90 isosceles right triangle, the two legs are equal, making their ratio exactly 1.
The 6 trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). The first three are primary functions; the last three are their reciprocals.
| Function | At 30° | At 45° | At 60° |
|---|---|---|---|
| sin | 0.5 | 0.7071 | 0.8660 |
| cos | 0.8660 | 0.7071 | 0.5 |
| tan | 0.5774 | 1 | 1.7321 |
| csc | 2 | 1.4142 | 1.1547 |
SOH-CAH-TOA is a mnemonic for remembering the three primary trig ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. It works for any right triangle.
| Mnemonic | Formula | Example (3-4-5) |
|---|---|---|
| SOH | sin = opp/hyp | sin(A) = 3/5 = 0.6 |
| CAH | cos = adj/hyp | cos(A) = 4/5 = 0.8 |
| TOA | tan = opp/adj | tan(A) = 3/4 = 0.75 |
Degrees and radians are two units for measuring angles. A full circle is 360 degrees or 2π radians. To convert: radians = degrees × π/180. Radians are preferred in calculus and physics because they simplify many formulas.
| Degrees | Radians | Fraction of π |
|---|---|---|
| 0° | 0 | 0 |
| 30° | 0.5236 | π/6 |
| 45° | 0.7854 | π/4 |
| 90° | 1.5708 | π/2 |
Inverse trig functions (arcsin, arccos, arctan) work backwards from a trig value to find the angle. For example, if sin(θ) = 0.5, then arcsin(0.5) = 30°. Arcsin and arccos accept inputs from -1 to 1; arctan accepts any real number.
| Function | Input | Output (degrees) |
|---|---|---|
| arcsin | 0.5 | 30° |
| arccos | 0.5 | 60° |
| arctan | 1 | 45° |
| arctan | 0 | 0° |
The unit circle is a circle with radius 1 centered at the origin. Any point (x, y) on it satisfies x = cos(θ) and y = sin(θ). It provides exact values for all standard angles and is the foundation of trigonometry.
| Angle | cos (x) | sin (y) |
|---|---|---|
| 0° | 1 | 0 |
| 90° | 0 | 1 |
| 180° | -1 | 0 |
| 270° | 0 | -1 |
Trigonometry studies the relationships between angles and sides of triangles. The six trigonometric functions — sine, cosine, tangent, cosecant, secant, and cotangent — form the foundation of this branch of mathematics and appear throughout science, engineering, and everyday applications.
The mnemonic SOH-CAH-TOA helps students remember the three primary ratios. In a right triangle, sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent. These ratios remain constant for any given angle regardless of triangle size.
Our trigonometry calculator computes all six functions simultaneously for any angle in degrees or radians. It also supports inverse trig functions to find angles from known values, displays reference angles and quadrant information, and provides a common-values reference table for quick lookup.
Last Updated: Mar 9, 2026
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