What are my trigonometry values?
Type an angle in degrees or radians to get all six trig functions, with exact values for special angles. Or pick Two sides to use SOH CAH TOA. The trigonometry calculator works as you type.
- sin θ
- 0.5
For θ = 30°, sin θ = 0.5 and cos θ = 0.866025.
- cos θ
- 0.866025
- tan θ
- 0.57735
- csc θ
- 2
- sec θ
- 1.154701
- cot θ
- 1.732051
- Exact values
- sin = 1/2, cos = √3/2, tan = √3/3, csc = 2, sec = 2√3/3, cot = √3
- Angle θ
- 30°
- Reference angle
- 30°
- Coterminal angle
- 30°
- Quadrant
- Quadrant I
sin θ: 0.5. For θ = 30°, sin θ = 0.5 and cos θ = 0.866025.
How to calculate
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).
Example with the default inputs (Start from An angle, Angle θ 30 °): For θ = 30°, sin θ = 0.5 and cos θ = 0.866025.
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.
- 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
Each example is checked against the calculator on every build.
- Start from An angle, Angle θ 30° gives sin θ 0.5, cos θ 0.866025, tan θ 0.57735, csc θ 2, sec θ 1.154701, cot θ 1.732051, Exact values sin = 1/2, cos = √3/2, tan = √3/3, csc = 2, sec = 2√3/3, cot = √3, Quadrant Quadrant I.
- Start from An angle, Angle θ 135° gives sin θ 0.707107, cos θ -0.707107, tan θ -1, Exact values sin = √2/2, cos = -√2/2, tan = -1, csc = √2, sec = -√2, cot = -1, Reference angle 45°, Quadrant Quadrant II.
- Start from An angle, Angle θ 90° gives sin θ 1, cos θ 0, csc θ 1, cot θ 0, Exact values sin = 1, cos = 0, tan = undefined, csc = 1, sec = undefined, cot = 0, Quadrant On the positive y-axis.
- Start from An angle, Angle θ 57.3° gives sin θ 0.841471, cos θ 0.540302, tan θ 1.557408, cot θ 0.642093, Coterminal angle 57.29578°, Quadrant Quadrant I.Source: NIST DLMF §4.14
- Start from An angle, Angle θ -420° gives sin θ -0.866025, cos θ 0.5, Exact values sin = -√3/2, cos = 1/2, tan = -√3, csc = -2√3/3, sec = 2, cot = -√3/3, Coterminal angle 300°, Reference angle 60°, Quadrant Quadrant IV.
- Start from Two sides, Which two sides do you know? Opposite and adjacent, Opposite side 3, Adjacent side 4 gives Hypotenuse 5, sin θ 0.6, cos θ 0.8, tan θ 0.75, Angle θ 36.869898°, Exact values sin = 3/5, cos = 4/5, tan = 3/4, csc = 5/3, sec = 5/4, cot = 4/3.
- Start from Two sides, Which two sides do you know? Opposite and hypotenuse, Opposite side 1, Hypotenuse 2 gives Adjacent side 1.732051, sin θ 0.5, Angle θ 30°.
How it works
From an angle
On the unit circle (radius 1, centre at the origin), the angle θ is measured from the positive x-axis, counterclockwise for positive angles. The point where it ends is (cos θ, sin θ). Then:
- sin θ = y, cos θ = x
- tan θ = sin θ ÷ cos θ
- csc θ = 1 ÷ sin θ, sec θ = 1 ÷ cos θ, cot θ = 1 ÷ tan θ = cos θ ÷ sin θ
The page also shows:
- Coterminal angle: θ plus or minus whole turns, from 0° up to but not including 360°.
- Quadrant: Quadrant I (0° to 90°), II (90° to 180°), III (180° to 270°), or IV (270° to 360°), by the coterminal angle. Exactly 0°, 90°, 180°, and 270° are on an axis: "On the positive x-axis", "On the positive y-axis", "On the negative x-axis", "On the negative y-axis".
- Reference angle: for a coterminal angle c, it is c in quadrant I, 180° − c in quadrant II, c − 180° in quadrant III, and 360° − c in quadrant IV.
Exact values. An angle within one part in 10¹² of a multiple of 15° (as a fraction of the angle's size in radians, or of 1 radian for angles under 1 radian) counts as exactly that multiple. For these angles the six values come from this table, with the signs of the quadrant (sin and csc are positive in quadrants I and II, cos and sec in I and IV, tan and cot in I and III):
| θ | sin | cos | tan | csc | sec | cot |
|---|---|---|---|---|---|---|
| 0° | 0 | 1 | 0 | undefined | 1 | undefined |
| 15° | (√6 − √2)/4 | (√6 + √2)/4 | 2 − √3 | √6 + √2 | √6 − √2 | 2 + √3 |
| 30° | 1/2 | √3/2 | √3/3 | 2 | 2√3/3 | √3 |
| 45° | √2/2 | √2/2 | 1 | √2 | √2 | 1 |
| 60° | √3/2 | 1/2 | √3 | 2√3/3 | 2 | √3/3 |
| 75° | (√6 + √2)/4 | (√6 − √2)/4 | 2 + √3 | √6 − √2 | √6 + √2 | 2 − √3 |
| 90° | 1 | 0 | undefined | 1 | undefined | 0 |
In quadrants II to IV, an angle takes the values of its reference angle. The exact values are written as sin = …, cos = …, tan = …, csc = …, sec = …, cot = …, a negative value with a leading - (a sum in brackets, such as -(2 + √3)), and undefined where the function has no value. The numbers shown are the same exact values rounded, so sin 180° is 0, not the tiny 1.2 × 10⁻¹⁶ that a rounded π would give. Where a function is undefined, it shows no number.
Any other angle uses the standard sine, cosine, and tangent of its value in radians, and has no exact values.
From two sides (SOH CAH TOA)
In a right triangle, the opposite side is across from θ, the adjacent side is next to θ, and the hypotenuse is across from the right angle. Pick which two sides you know (opposite and adjacent, opposite and hypotenuse, or adjacent and hypotenuse) and type them:
- The third side by the Pythagorean theorem: hypotenuse = √(opposite² + adjacent²), or a leg = √(hypotenuse² − other leg²). The sides are read exactly as their decimals and the sum of squares is exact, so only the square root is rounded.
- sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent, csc θ = hypotenuse ÷ opposite, sec θ = hypotenuse ÷ adjacent, cot θ = adjacent ÷ opposite.
- θ = arctan(opposite ÷ adjacent), computed as atan2(opposite, adjacent).
When all three sides are exact fractions (the missing side's square is a fraction whose top and bottom are both perfect squares, as with 3, 4, 5), the six ratios also show as exact fractions in lowest terms, such as sin = 3/5, cos = 4/5, tan = 3/4, csc = 5/3, sec = 5/4, cot = 4/3.
Rules and messages
- The hypotenuse must be longer than each other side: "The hypotenuse must be longer than each of the other two sides."
- A triangle with a value beyond what a 64-bit float can hold gives no answer: "The triangle is too large or too small to work out."
- An angle can be any value from −1,000,000 to 1,000,000 radians (about ±57 million degrees), typed in degrees or radians.
Assumptions
- Angles show in degrees. Values show with at most 6 decimals, or 6 significant figures below 0.0001, rounded half up on the decimal value.
Worked examples by hand
θ = 30°. The point on the unit circle is (√3/2, 1/2). So sin 30° = 1/2 = 0.5, cos 30° = √3/2 = 0.866025, tan 30° = (1/2) ÷ (√3/2) = 1/√3 = √3/3 = 0.57735, csc 30° = 2, sec 30° = 2√3/3 = 1.154701, cot 30° = √3 = 1.732051. Quadrant I.
θ = 135°. Quadrant II, reference angle 180° − 135° = 45°. sin 135° = √2/2, cos 135° = −√2/2, tan 135° = −1.
θ = 90°. The point is (0, 1): sin 90° = 1, cos 90° = 0, and tan 90° = 1 ÷ 0 is undefined, as is sec 90°. On the positive y-axis.
θ = 1 radian (57.2958°). sin 1 = 0.841471, cos 1 = 0.540302, tan 1 = 1.557408, cot 1 = 0.642093. Not a multiple of 15°, so there are no exact values.
θ = −420°. Adding two turns: −420° + 720° = 300°, in quadrant IV with reference angle 360° − 300° = 60°. sin(−420°) = −√3/2, cos(−420°) = 1/2, tan(−420°) = −√3.
Opposite 3, adjacent 4. Hypotenuse = √(9 + 16) = 5. sin θ = 3/5, cos θ = 4/5, tan θ = 3/4, and θ = arctan(3/4) = 36.869898°.
Opposite 1, hypotenuse 2. Adjacent = √(4 − 1) = √3 = 1.732051. sin θ = 1/2, so θ = 30°. The adjacent side is not a fraction, so the ratios show as decimals only.
Other questions people ask
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°.