The formula
sin = opposite ÷ hypotenuse · cos = adjacent ÷ hypotenuse · tan = sin ÷ cos
Where it comes from
In a right triangle the ratios between sides depend only on the angles, not on the size. That single fact is what makes trigonometry work: scale the triangle up and every ratio stays put, so each angle has its own fixed sine, cosine and tangent.
How to work it out by hand
- Convert the angle to radians if your calculator needs them
- Sine is the opposite side over the hypotenuse
- Cosine is the adjacent side over the hypotenuse
- Tangent is sine divided by cosine
What is worth knowing
The identity sin² + cos² = 1 is Pythagoras in disguise: on the unit circle the sine and cosine are the two legs and the radius is 1. It is the fastest check there is on any trigonometric answer. The tangent is undefined at 90° and 270° because the cosine is zero there and you cannot divide by it — not because the value is 'infinite', which is loose talk. The other three ratios are just reciprocals: secant is 1/cos, cosecant is 1/sin, cotangent is 1/tan, and they exist mostly because they were easier to look up in tables before calculators.