Trig Ratios and the Unit Circle

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.

For a point $P(x,y)$ at distance $r>0$ from the origin:

Here $x,y$ are signed coordinates, not negative side lengths. Left gives negative cosine; below gives negative sine. Tangent divides these two coordinates, so equal signs give positive tangent.

A unit circle has radius 1. Its point is therefore $(\cos\theta,\sin\theta)$. Move the angle and compare the point with the two graphs.

The circle cannot go farther than 1 in either coordinate, so sine and cosine stay between −1 and 1. At $90^\circ$ and $270^\circ$, cosine is zero: tangent is undefined because division by zero is not allowed.

A full turn is $360^\circ=2\pi$ radians. Adding or subtracting a full turn returns to the same point and gives the same ratios.