# What are my trigonometry values?

Finds sin, cos, tan, csc, sec, and cot of any angle in degrees or radians, with exact values for multiples of 15°, or the ratios and angle of a right triangle from two sides (SOH CAH TOA).

- Page: https://www.acalculator.org/math/trigonometry-calculator
- JSON spec: https://www.acalculator.org/math/trigonometry-calculator.json
- Version: 85579f61736f

## Default answer

Example with the default inputs (Start from An angle, Angle θ 30 °): For θ = 30°, sin θ = 0.5 and cos θ = 0.866025.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| mode | Start from | An angle, or two sides of a right triangle. |
| theta | Angle θ | Any angle, in degrees or radians; negative angles turn clockwise. |
| pair | Which two sides do you know? | The two sides of the right triangle you know; the third is found. |
| opp | Opposite side | The side of the right triangle across from angle θ. |
| adj | Adjacent side | The side next to angle θ that is not the hypotenuse. |
| hyp | Hypotenuse | The longest side, across from the right angle. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| sin | sin θ | Sine: opposite ÷ hypotenuse, the y value on the unit circle. |
| cos | cos θ | Cosine: adjacent ÷ hypotenuse, the x value on the unit circle. |
| tan | tan θ | Tangent: opposite ÷ adjacent = sin ÷ cos. Undefined at 90° and 270°. |
| csc | csc θ | Cosecant: 1 ÷ sin. Undefined at 0° and 180°. |
| sec | sec θ | Secant: 1 ÷ cos. Undefined at 90° and 270°. |
| cot | cot θ | Cotangent: 1 ÷ tan = cos ÷ sin. Undefined at 0° and 180°. |
| exact | Exact values | The six values as exact fractions and square roots, for multiples of 15° or whole-number sides. |
| angle | Angle θ | The angle: as typed, or from the two sides (θ = arctan(opposite ÷ adjacent)). |
| reference | Reference angle | The acute angle between θ and the x-axis, from 0° to 90°. |
| coterminal | Coterminal angle | The same direction as θ, from 0° up to 360°. |
| quadrant | Quadrant | Where θ ends on the unit circle. |
| opposite | Opposite side | The side across from θ. |
| adjacent | Adjacent side | The side next to θ. |
| hypotenuse | Hypotenuse | The side across from the right angle. |

## Method

On the unit circle, sin θ = y and cos θ = x; tan θ = sin θ ÷ cos θ, csc θ = 1 ÷ sin θ, sec θ = 1 ÷ cos θ, cot θ = 1 ÷ tan θ. In a right triangle, SOH CAH TOA: sin = opposite ÷ hypotenuse, cos = adjacent ÷ hypotenuse, tan = opposite ÷ adjacent.

## Assumptions

- Angles can be typed in degrees or radians, and can be negative or more than 360°.
- An angle within one part in 10¹² of a multiple of 15° is taken as that multiple, so the values there are exact (sin 180° = 0, tan 90° undefined).
- A function that is undefined at the angle (tan 90°, cot 0°) shows no value.

## Worked examples

1. mode = angle, theta = 0.523599 gives sin = 0.5, cos = 0.866025, tan = 0.57735, csc = 2, sec = 1.154701, cot = 1.732051, exact = sin = 1/2, cos = √3/2, tan = √3/3, csc = 2, sec = 2√3/3, cot = √3, quadrant = Quadrant I.
2. mode = angle, theta = 2.356194 gives sin = 0.707107, cos = -0.707107, tan = -1, exact = sin = √2/2, cos = -√2/2, tan = -1, csc = √2, sec = -√2, cot = -1, reference = 0.785398, quadrant = Quadrant II.
3. mode = angle, theta = 1.570796 gives sin = 1, cos = 0, csc = 1, cot = 0, exact = sin = 1, cos = 0, tan = undefined, csc = 1, sec = undefined, cot = 0, quadrant = On the positive y-axis.
4. mode = angle, theta = 1 gives sin = 0.841471, cos = 0.540302, tan = 1.557408, cot = 0.642093, coterminal = 1, quadrant = Quadrant I. Source: NIST DLMF §4.14.
5. mode = angle, theta = -7.330383 gives sin = -0.866025, cos = 0.5, exact = sin = -√3/2, cos = 1/2, tan = -√3, csc = -2√3/3, sec = 2, cot = -√3/3, coterminal = 5.235988, reference = 1.047198, quadrant = Quadrant IV.
6. mode = sides, pair = oa, opp = 3, adj = 4 gives hypotenuse = 5, sin = 0.6, cos = 0.8, tan = 0.75, angle = 0.643501, exact = sin = 3/5, cos = 4/5, tan = 3/4, csc = 5/3, sec = 5/4, cot = 4/3.
7. mode = sides, pair = oh, opp = 1, hyp = 2 gives adjacent = 1.732051, sin = 0.5, angle = 0.523599.

## FAQ

### What does SOH CAH TOA mean?

It is a memory aid for the ratios in a right triangle: Sine = Opposite ÷ Hypotenuse, Cosine = Adjacent ÷ Hypotenuse, Tangent = Opposite ÷ Adjacent. In a 3-4-5 triangle with the angle across from the side 3, sin = 3/5, cos = 4/5, and tan = 3/4.

### What are sin, cos, and tan of 30°, 45°, and 60°?

sin 30° = 1/2, cos 30° = √3/2, tan 30° = √3/3. sin 45° = cos 45° = √2/2, tan 45° = 1. sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3. The calculator shows exact values like these for every multiple of 15°.

### Why is tan 90° undefined?

tan θ = sin θ ÷ cos θ, and cos 90° = 0, so tan 90° would divide by zero. On the unit circle, the point at 90° is (0, 1), straight up, where the tangent line never meets the x-axis. The same is true at 270°. cot and csc are undefined at 0° and 180°, where sin is 0.

### What are csc, sec, and cot?

They are the reciprocals: cosecant csc θ = 1 ÷ sin θ, secant sec θ = 1 ÷ cos θ, and cotangent cot θ = 1 ÷ tan θ. For 60°, sec 60° = 1 ÷ (1/2) = 2.

### How do I use degrees and radians?

Pick the unit next to the angle box. A full turn is 360° or 2π radians, so radians = degrees × π ÷ 180. 90° is π/2 ≈ 1.5708 radians; 1 radian is about 57.2958°.

### What is a reference angle?

The acute angle between the end of θ and the x-axis. Trig values of θ equal those of its reference angle, up to the sign set by the quadrant. 135° has a reference angle of 45°, so sin 135° = sin 45° = √2/2 and cos 135° = −√2/2.

### How do I find an angle from two sides?

Pick "Two sides". The angle θ across from the opposite side is arctan(opposite ÷ adjacent). With opposite 3 and adjacent 4, θ = arctan(0.75) = 36.87°. With opposite 1 and hypotenuse 2, sin θ = 1/2, so θ = 30°.

## Sources

- OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (SOH CAH TOA, reciprocal functions) and §7.3 Unit Circle (special angles, reference angles, quadrants). https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle
- OpenStax, Algebra and Trigonometry 2e, §9.2 Sum and Difference Identities (sin 15° = (√6 − √2)/4, tan 15° = 2 − √3). https://openstax.org/books/algebra-and-trigonometry-2e/pages/9-2-sum-and-difference-identities
- NIST Digital Library of Mathematical Functions, §4.14 Definitions and Periodicity (trigonometric functions). https://dlmf.nist.gov/4.14