Trigonometric Functions Calculator
Find sin, cos, tan, cot, sec, and csc for any angle in degrees or radians instantly.
Calculate Trigonometric Functions
| Function | Formula | Value |
|---|
Common Angle Reference Table
Common angle reference values are used constantly in mathematics, physics, and engineering. The table below lists exact and decimal values for all six trigonometric functions at the most frequently used angles.
| Degrees | Radians | sin | cos | tan | cot | sec | csc |
|---|---|---|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 | undef. | 1 | undef. |
| 30° | π/6 | 0.5 | 0.8660 | 0.5774 | 1.7321 | 1.1547 | 2 |
| 45° | π/4 | 0.7071 | 0.7071 | 1 | 1 | 1.4142 | 1.4142 |
| 60° | π/3 | 0.8660 | 0.5 | 1.7321 | 0.5774 | 2 | 1.1547 |
| 90° | π/2 | 1 | 0 | undef. | 0 | undef. | 1 |
| 120° | 2π/3 | 0.8660 | −0.5 | −1.7321 | −0.5774 | −2 | 1.1547 |
| 135° | 3π/4 | 0.7071 | −0.7071 | −1 | −1 | −1.4142 | 1.4142 |
| 150° | 5π/6 | 0.5 | −0.8660 | −0.5774 | −1.7321 | −1.1547 | 2 |
| 180° | π | 0 | −1 | 0 | undef. | −1 | undef. |
| 270° | 3π/2 | −1 | 0 | undef. | 0 | undef. | −1 |
| 360° | 2π | 0 | 1 | 0 | undef. | 1 | undef. |
What Are Trigonometric Functions?
Trigonometric functions are mathematical functions that relate angles of a right triangle to the ratios of its sides. They are also called circular functions because they can be defined using the unit circle, where the radius equals 1. These six functions form the foundation of trigonometry, a branch of mathematics with applications in physics, engineering, architecture, navigation, and computer graphics.
The six trigonometric functions all take an angle as input and return a dimensionless ratio. For a right triangle with angle θ, the sides are labeled as opposite (the side across from the angle), adjacent (the side next to the angle), and hypotenuse (the longest side, across from the 90° angle).
Sine Function (sin)
Sine equals the ratio of the opposite side to the hypotenuse: sin(θ) = opposite / hypotenuse. The sine function has a range from −1 to 1 and a period of 360 degrees (2π radians). It starts at 0, peaks at 1 at 90°, returns to 0 at 180°, reaches −1 at 270°, and completes its cycle at 360°.
Cosine Function (cos)
Cosine equals the ratio of the adjacent side to the hypotenuse: cos(θ) = adjacent / hypotenuse. Like sine, cosine has a range of −1 to 1 and a period of 360°. Cosine equals 1 at 0°, drops to 0 at 90°, reaches −1 at 180°, returns to 0 at 270°, and completes the cycle at 360°.
Tangent Function (tan)
Tangent equals the ratio of the opposite side to the adjacent side: tan(θ) = opposite / adjacent. Equivalently, tan(θ) = sin(θ) / cos(θ). Tangent has an unlimited range (negative infinity to positive infinity) and a period of 180°. It is undefined wherever cosine equals zero, at 90°, 270°, and all odd multiples of 90°.
Cotangent, Secant, and Cosecant
Cotangent (cot), secant (sec), and cosecant (csc) are the reciprocal functions. Cotangent equals 1 / tan(θ) or cos(θ) / sin(θ). Secant equals 1 / cos(θ). Cosecant equals 1 / sin(θ). Each reciprocal function is undefined wherever its denominator function equals zero.
How to Use This Trig Calculator
How to use this trigonometric functions calculator is straightforward. Enter your angle value in the first field. Choose whether your angle is measured in degrees or radians. Select the function you want to calculate, or choose “All Six Functions” to see every result at once. Press Calculate or hit Enter. The results table shows the function name, its formula, and the computed value. Press Clear to reset all fields and start over.
Degrees vs. Radians
Degrees and radians are two systems for measuring angles. A full rotation is 360 degrees or 2π radians. Most everyday problems use degrees, while calculus, physics equations, and programming languages typically use radians. To convert between them: multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees.
Reading the Results
Each result shows the function name, the defining formula based on triangle sides, and the numeric value rounded to 8 significant figures. Very large or very small values appear in scientific notation. When a function is undefined at your angle (for example, tan(90°)), the result displays “Undefined” rather than a number.
Trigonometric Identities and Formulas
Trigonometric identities are equations involving trig functions that hold true for all values of the variable. Knowing these identities lets you simplify complex expressions and solve equations that would otherwise be intractable.
Pythagorean Identities
The most fundamental identity states that sin²(θ) + cos²(θ) = 1 for any angle θ. Dividing each term by cos²(θ) gives 1 + tan²(θ) = sec²(θ). Dividing by sin²(θ) gives cot²(θ) + 1 = csc²(θ). All three of these come directly from the Pythagorean theorem applied to the unit circle.
Angle Sum and Difference Formulas
Angle addition formulas allow you to compute trig values for compound angles. The sine of a sum is sin(A + B) = sin A cos B + cos A sin B. The cosine of a sum is cos(A + B) = cos A cos B − sin A sin B. The tangent of a sum is tan(A + B) = (tan A + tan B) / (1 − tan A tan B). These formulas are the building blocks for double-angle and half-angle identities.
Reciprocal and Quotient Identities
The six trig functions relate to each other through reciprocal and quotient relationships: csc(θ) = 1/sin(θ), sec(θ) = 1/cos(θ), cot(θ) = 1/tan(θ), tan(θ) = sin(θ)/cos(θ), and cot(θ) = cos(θ)/sin(θ). These relationships make it possible to express any of the six functions in terms of any other.
Frequently Asked Questions
The six trigonometric functions are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). They relate angles of a right triangle to the ratios of its sides, and are also definable using the unit circle.
To convert degrees to radians, multiply the degree value by π divided by 180. For example, 90 degrees equals 90 × (π/180) = π/2 ≈ 1.5708 radians. To go the other direction, multiply radians by 180/π.
The value of sin 30 degrees is exactly 0.5. This is one of the most commonly used exact trigonometric values. It comes from the ratio of the opposite side (1) to the hypotenuse (2) in a 30-60-90 right triangle.
Secant (sec) is the reciprocal of cosine. It equals 1 divided by cos(θ). Secant is undefined wherever cosine equals zero, which occurs at 90° and 270° (or π/2 and 3π/2 radians). The secant function has no value between −1 and 1.
Tangent is undefined at 90 degrees (π/2 radians), at 270 degrees (3π/2 radians), and at any angle of the form 90 + 180n degrees, where n is any integer. At these angles, cosine equals zero, and division by zero makes the function undefined.
Cos 45 degrees equals √2 / 2, which is approximately 0.7071. At 45 degrees, the sine and cosine values are equal, which is why sin 45° and cos 45° are both √2/2. This angle corresponds to an isosceles right triangle.
Sine (sin) and cosecant (csc) are reciprocals of each other. Csc(θ) = 1 / sin(θ). When sin(θ) = 0.5, csc(θ) = 2. Cosecant is undefined whenever sine equals zero, which is at 0°, 180°, 360°, and their multiples.