Trig Ratios and the Unit Circle

Sine, cosine and tangent compare lengths in a right-angled triangle. A circle lets us use the same idea for angles beyond $90^\circ$.

Choose the angle $\theta$ first. The hypotenuse is opposite the right angle. The opposite side faces $\theta$; the adjacent side touches it but is not the hypotenuse.

These are ratios, so they have no units. Enlarging the triangle multiplies both lengths by the same amount and leaves each ratio unchanged. If you choose the other acute angle, opposite and adjacent swap.

For the angle shown, sine is $3/5$, cosine is $4/5$ and tangent is $3/4$. To find an opposite side when the hypotenuse is 10:

Start along the positive horizontal axis. A positive angle turns anticlockwise; a negative angle turns clockwise. The axes divide the plane into four quadrants, numbered I, II, III, IV anticlockwise from the top right.