Values for common angles
| 角度Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 = 0.5000 | √3/2 ≈ 0.8660 | √3/3 ≈ 0.5774 |
| 45° | √2/2 ≈ 0.7071 | √2/2 ≈ 0.7071 | 1 |
| 60° | √3/2 ≈ 0.8660 | 1/2 = 0.5000 | √3 ≈ 1.7321 |
| 90° | 1 | 0 | 无定义Undefined |
| 120° | √3/2 ≈ 0.8660 | -1/2 = -0.5000 | -√3 ≈ -1.7321 |
| 135° | √2/2 ≈ 0.7071 | -√2/2 ≈ -0.7071 | -1 |
| 150° | 1/2 = 0.5000 | -√3/2 ≈ -0.8660 | -√3/3 ≈ -0.5774 |
| 180° | 0 | -1 | 0 |
A memory trick: the sines of 30°, 45° and 60° are √1/2, √2/2 and √3/2, that is, the number under the root counting 1, 2, 3; the cosines simply run in reverse (cos 60° equals sin 30°). In the second quadrant (90°–180°) only sine stays positive while cosine and tangent turn negative, which is exactly why the reduction formulas have to handle signs.
Degrees and radians
| 角度Degrees | 弧度Radians | 近似值Approx. |
|---|---|---|
| 0° | 0 | 0 |
| 30° | π/6 | 0.5236 |
| 45° | π/4 | 0.7854 |
| 60° | π/3 | 1.0472 |
| 90° | π/2 | 1.5708 |
| 180° | π | 3.1416 |
| 270° | 3π/2 | 4.7124 |
| 360° | 2π | 6.2832 |
To convert: 1 radian ≈ 57.2958°, and 1° = π/180 ≈ 0.0174533 radians. Radians are the default because they make formulas such as arc length = radius × angle and sector area = ½r²θ as clean as possible; in calculus, only radians give the tidy (sin x)′ = cos x. In code, remember that most standard libraries take radians: passing degrees is a classic bug (tan(45) returns 1.6198 because it means 45 radians, roughly 2,578°).
Identities and reduction formulas
| 名称Name | 公式Formula |
|---|---|
| 平方关系Pythagorean | sin²θ + cos²θ = 1 |
| 商数关系Quotient | tanθ = sinθ / cosθ |
| 余角关系Co-function | sin(90° − θ) = cosθ |
| 补角关系Supplementary | sin(180° − θ) = sinθ，cos(180° − θ) = −cosθ |
| 和角公式Angle sum | sin(α ± β) = sinα·cosβ ± cosα·sinβ |
| 差角余弦Cosine difference | cos(α ± β) = cosα·cosβ ∓ sinα·sinβ |
| 倍角公式Double angle | sin2θ = 2sinθ·cosθ，cos2θ = cos²θ − sin²θ |
| 正弦定理Law of sines | a / sinA = b / sinB = c / sinC = 2R |
| 余弦定理Law of cosines | c² = a² + b² − 2ab·cosC |
Domain and periodicity
Sine and cosine are defined on all real numbers with values in [-1, 1] and a period of 2π; tangent has period π but is undefined at 90°, 270° and generally at π/2 + kπ, because there cosθ = 0 and tanθ = sinθ / cosθ becomes a division by zero — the limit runs to ±∞, so the answer is not "a very large number" but "no value". Mind the sign conventions too: in physics angles are usually measured anticlockwise from the positive x-axis, while bearings in surveying and navigation are measured clockwise from north — a 90° difference in both origin and direction, so settle the convention before substituting.
Where you actually use it
Three common settings. Slopes: gradient i = tanθ = rise ÷ run, so a 5% grade is about 2.86° and 45° is 100%. Solving triangles: use the law of cosines when you know two sides and the included angle, and the law of sines when you know two angles and a side — the basis of surveying, measurement and setting out on site. Solid geometry and vectors: angles between planes and distances from a point to a plane all reduce to right triangles with sine and cosine. For actual numbers, try the site’s <a href="solid.html">solid geometry tool</a> (with 3D visuals) or convert units with the <a href="unit.html">unit converter</a>.
FAQ
Why is sin 30° equal to 0.5?
Place 30° on the unit circle and the y-coordinate of the point is sin θ. For 30° the hypotenuse is twice the short leg (splitting an equilateral triangle in half), so sin 30° = 1/2 = 0.5.
What is tan 90°?
Undefined. Since tanθ = sinθ / cosθ and cos 90° = 0, the expression divides by zero; approaching from either side tends to +∞ and −∞, so it is not "infinity" either.
Why does sin(45) give the wrong answer in code?
Because standard libraries take radians: sin(45) computes the sine of 45 radians (about 2,578°). Convert with degrees × π ÷ 180, or use the language’s radians helper.