In more detail
In a right triangle sin = opposite / hypotenuse, cos = adjacent / hypotenuse and tan = opposite / adjacent. In a 3-4-5 triangle the angle opposite the side 4 is atan(4/3) = 53.13°. The inverse functions (asin, acos, atan) turn a ratio back into an angle. For triangles without a right angle, the law of sines a / sin A = b / sin B and the law of cosines c² = a² + b² − 2ab × cos C apply. On the unit circle, sin and cos are the y and x coordinates, so sin² θ + cos² θ = 1.
Why it matters
Trigonometry gives heights and distances you cannot measure directly, slopes of ramps and roofs, and the components of forces and velocities. It also describes anything periodic, from sound waves to seasons. The classic error is the angle unit: spreadsheets and programming languages work in radians, so sin(30) means 30 radians. Check your calculator is in degree or radian mode as intended.
Example
ramp 10 m long at 5° rise = 10 × sin 5° = 0.872 m
tree 50 m away, top at 30° height = 50 × tan 30° = 28.87 m
sides 7 and 9 with 60° between them
third side = √(49 + 81 − 2 × 7 × 9 × cos 60°) = √67 = 8.185
>>> import math
>>> math.sin(30), math.sin(math.radians(30))
(-0.9880316240928618, 0.49999999999999994)
Common confusion
The Pythagorean theorem is the law of cosines with C = 90°, where cos C = 0 and the last term disappears.