Trigonometry Reference
The tables and identities worth having to hand, plus the detail most references omit - what range each inverse function actually returns.
Last updated: August 2026 · Written and verified by Akash Pandey
The three ratios
In a right-angled triangle, each ratio compares two sides relative to one of the acute angles. SOH-CAH-TOA is the usual mnemonic.
sin θ = opposite / hypotenuse cos θ = adjacent / hypotenuse tan θ = opposite / adjacent
- Since
tan θ = sin θ / cos θ, the third follows from the first two. - On the unit circle - radius 1, centred at the origin - a point at angle θ has coordinates exactly
(cos θ, sin θ).
Exact values
| Degrees | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undefined |
| 180° | π | 0 | −1 | 0 |
| 270° | 3π/2 | −1 | 0 | undefined |
| 360° | 2π | 0 | 1 | 0 |
The 30-60-90 and 45-45-90 values are worth memorising because they come from two triangles you can reconstruct: half an equilateral triangle of side 2 gives the first set, and a unit square cut along its diagonal gives the second.
Signs by quadrant
| Quadrant | Angle range | Positive there |
|---|---|---|
| I | 0°–90° | All three |
| II | 90°–180° | Sine only |
| III | 180°–270° | Tangent only |
| IV | 270°–360° | Cosine only |
Core identities
| Identity | Statement |
|---|---|
| Pythagorean | sin²θ + cos²θ = 1 |
| Pythagorean (tan) | 1 + tan²θ = sec²θ |
| Quotient | tan θ = sin θ / cos θ |
| Sine double angle | sin 2θ = 2 sin θ cos θ |
| Cosine double angle | cos 2θ = cos²θ − sin²θ |
| Sine addition | sin(A+B) = sin A cos B + cos A sin B |
| Cosine addition | cos(A+B) = cos A cos B − sin A sin B |
| Even / odd | cos(−θ) = cos θ, sin(−θ) = −sin θ |
The Pythagorean identity is just Pythagoras' theorem applied to the unit circle: the point (cos θ, sin θ) is at distance 1 from the origin, so cos²θ + sin²θ = 1².
The inverse functions and their ranges
This is where most references stop short. Sine takes the value 0.5 at 30°, but also at 150°, 390°, and infinitely many other angles. For asin to be a function it must return exactly one of them, so each inverse is restricted to a principal range:
| Function | Accepts | Returns (degrees) | Returns (radians) |
|---|---|---|---|
| asin(x) | −1 ≤ x ≤ 1 | −90° to 90° | −π/2 to π/2 |
| acos(x) | −1 ≤ x ≤ 1 | 0° to 180° | 0 to π |
| atan(x) | any real number | −90° to 90° | −π/2 to π/2 |
Frequently Asked Questions
What is the exact value of sin(45°)?
√2/2, which is approximately 0.707107. It is sometimes written as 1/√2 - the same number, rationalised differently.
Why is tan(90°) undefined?
Because tan θ = sin θ / cos θ and cos(90°) = 0, so it would require dividing by zero. In floating point the calculator returns an enormous finite number instead, for reasons explained in the degrees vs radians guide.
What does SOH-CAH-TOA stand for?
Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
Why does asin give me the wrong angle?
It gives the angle in its principal range of −90° to 90°. If the angle you want is obtuse, compute 180° − the returned value.
What is the difference between arcsin and sin⁻¹?
Nothing - they are two notations for the same inverse function. Note that sin⁻¹(x) does not mean 1/sin(x), which is a genuinely confusing piece of standard notation.
Do the identities work in both degrees and radians?
Yes, all of them. Only calculus results such as derivatives and Taylor series require radians specifically.