Common angle values
The table below lists exact sine, cosine and tangent values for the most frequently used angles, verified to six decimal places.
| Angle | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 0.5 | 0.866025 | 0.577350 |
| 45° | 0.707107 | 0.707107 | 1 |
| 60° | 0.866025 | 0.5 | 1.732051 |
| 90° | 1 | 0 | undefined |
- Tangent is undefined at 90° (and every 90° + 180°k) because cosine equals zero there, making the ratio sin/cos undefined.
- Mixing up the degrees/radians setting is the single most common trigonometry error — the same numeric value of 30 means a very different angle in degrees (30°) versus radians (about 1719°).
- For inverse functions, entering a value outside [−1, 1] for arcsine or arccosine is mathematically undefined, since no real angle has a sine or cosine outside that range.
What are the trigonometric functions?
Sine, cosine and tangent relate the angles of a right triangle to the ratios of its side lengths: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent (equivalently, sin θ / cos θ). Their inverse functions — arcsine, arccosine and arctangent — take a ratio as input and return the corresponding angle.
For sine and cosine, the two functions are complementary: sin(θ) = cos(90° − θ). This calculator displays that co-function value alongside the primary result for sin, cos, asin and acos, and shows the cotangent alongside tangent, so the relationship between related functions is visible at a glance.
How to use this trigonometry calculator
- Select the function: sin, cos, tan, or their inverses sin⁻¹, cos⁻¹, tan⁻¹.
- Select the angle unit: degrees or radians. This applies to angle inputs for forward functions and to the angle returned by inverse functions.
- Enter the value — an angle for sin/cos/tan, or a ratio for the inverse functions.
- Read the result and, where applicable, the related co-function value.
The trigonometric ratio formulas
The inverse functions asin and acos require an input between −1 and 1 inclusive, since sine and cosine values never fall outside that range.
Common mistakes
- Leaving the calculator in the wrong angle-unit mode (degrees vs. radians), which produces a completely different — but numerically valid-looking — result.
- Entering an angle for arcsine or arccosine input instead of a ratio between −1 and 1.
- Expecting tan(90°) to return a large number instead of "undefined" — mathematically the tangent function has a vertical asymptote there.
- Confusing a function with its inverse, such as using sin when cos⁻¹ was intended.
Часто задаваемые вопросы
What is sin(30°)?
sin(30°) = 0.5 exactly. Its co-function, cos(30°), equals √3/2 ≈ 0.866025, consistent with sin(30°) = cos(90° − 30°) = cos(60°).
What is cos(45°)?
cos(45°) = √2/2 ≈ 0.707107, which equals sin(45°) exactly, since 45° is its own complement (90° − 45° = 45°).
Why is tan(90°) undefined?
Tangent equals sine divided by cosine, and cos(90°) = 0. Division by zero is undefined, so tan(90°) has no finite value — the tangent function has a vertical asymptote at 90° and every 180° beyond it.
What is the difference between sin and sin⁻¹?
sin(θ) takes an angle and returns a ratio between −1 and 1. sin⁻¹(x), or arcsine, does the reverse: it takes a ratio between −1 and 1 and returns the corresponding angle.
How do you convert between degrees and radians?
Multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees. For example, 60° × π/180 ≈ 1.0472 radians.
What is arcsin(0.5)?
arcsin(0.5) = 30° (or π/6 radians), since sin(30°) = 0.5. The calculator also shows the complementary co-function angle, arccos(0.5) = 60°, since 30° + 60° = 90°.
Источники
- Weisstein, Eric W. "Trigonometric Functions." MathWorld — A Wolfram Web Resource.
- Standard trigonometry textbook conventions (e.g. Larson, Trigonometry, Cengage Learning).