Trigonometry Calculator
Calculate sin, cos, tan, csc, sec and cot for any angle in degrees or radians with visual unit circle indicator
Angle: 45.00° ★ Special angle
sin θ
0.707107
cos θ
0.707107
tan θ
1.000000
csc θ
1.414214
sec θ
1.414214
cot θ
1.000000
Unit Circle
Identities
sin²θ + cos²θ = 1
0.7071² + 0.7071² = 1.000000
SOH CAH TOA, Then the Reciprocals
In a right-angled triangle: sine of an angle is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent. SOH CAH TOA. The reciprocals are less famous but show up in A-level work: cosec (csc) = 1/sin, secant = 1/cos, cotangent = 1/tan. The calculator returns all six functions for any angle in degrees or radians.
Special angles are worth memorising because they come up constantly: sin 30° = 1/2, cos 60° = 1/2, sin 45° = cos 45° = √2/2, tan 45° = 1, sin 60° = √3/2. The calculator marks these as special when you hit one. For converting between degrees and radians directly, [Angle Converter](/angle-converter) is the dedicated tool, though this one switches between the two for input.
Where Tan Goes Undefined
tan(90°) and tan(270°) are undefined because cos(90°) = cos(270°) = 0, and tan = sin/cos. You will see 'undefined' rather than a number. Likewise, cosec is undefined at 0° and 180° (where sin is zero), and secant is undefined at 90° and 270° (where cos is zero). These are not bugs; they are mathematical realities the calculator surfaces explicitly so you do not get confused by an Infinity result.
An A-level question stuck on log identities is often better tackled in stages: simplify with identities first (sin² + cos² = 1, tan = sin/cos, double-angle formulas), then plug numbers in at the end. If the question asks for an exact value, leave answers as surds (√3/2, not 0.866). The calculator gives decimal values to six places, which is precise enough for any practical bearing, surveying, or engineering problem.
Common Angles
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 0.5 | 0.8660 | 0.5774 |
| 45° | 0.7071 | 0.7071 | 1 |
| 60° | 0.8660 | 0.5 | 1.7321 |
| 90° | 1 | 0 | undefined |
| 180° | 0 | -1 | 0 |
Frequently Asked Questions
What is the difference between degrees and radians?
Degrees split a full circle into 360 parts. Radians split it into 2π (about 6.28) parts. 180° = π radians. Radians are the natural unit for calculus and physics; degrees are more intuitive for everyday angles. Pick whichever your problem is set in and stay consistent.
Why is tan(90°) undefined?
Tan is sin divided by cos, and cos(90°) = 0. You cannot divide by zero. As an angle approaches 90°, tan grows without bound (tan(89.99°) ≈ 5,729). At exactly 90° it is undefined.
What does SOH CAH TOA mean?
It is the standard mnemonic for the three primary trig ratios. Sine = Opposite over Hypotenuse, Cosine = Adjacent over Hypotenuse, Tangent = Opposite over Adjacent. It only applies to right-angled triangles directly; for other triangles, use the sine rule or cosine rule.
How do I find an angle from a sine value?
Use inverse sine, written sin⁻¹ or arcsin. If sin(θ) = 0.5, then θ = sin⁻¹(0.5) = 30°. Most calculators have inverse buttons next to sin, cos, and tan. Be aware the inverse functions only return angles in a restricted range (sin⁻¹: -90° to 90°, cos⁻¹: 0° to 180°), so check whether the original problem expects a different one.