How Are Sine and Cosine Defined Using the Unit Circle?

Cosine and sine are defined as the coordinates of a point on the unit circle. If a radius makes an angle $\theta$ with the positive horizontal axis and meets the circle at $P$, then $P=(\cos\theta,\sin\theta)$.

This is AHL 3.8, so it is HL-only content in IB Mathematics: Applications and Interpretation.

How the definition works

The unit circle is the circle centred at the origin with radius $1$, so its equation is

$$x^2+y^2=1.$$

An angle $\theta$ is measured from the positive horizontal axis. Positive angles are measured anticlockwise. The terminal radius intersects the unit circle at a point $P=(x,y)$.

By definition,

$$\cos\theta=x \qquad \text{and} \qquad \sin\theta=y.$$

Therefore, cosine gives the point's horizontal coordinate, while sine gives its vertical coordinate. As the point moves around the circle, these coordinates explain why sine and cosine can be positive or negative.

The coordinate definition also explains periodicity. After one complete revolution, the point returns to the same coordinates, so both functions repeat every $2\pi$ radians. Circle symmetry also gives $\cos(-\theta)=\cos\theta$ and $\sin(-\theta)=-\sin\theta$.

For example, suppose $P$ lies in quadrant II and has $x=-0.6$. From the circle equation,

$$y=\sqrt{1-(-0.6)^2}=0.8.$$

The positive root is chosen because $y>0$ in quadrant II. Hence $\cos\theta=-0.6$ and $\sin\theta=0.8$; a GDC gives $\theta ={\cos }^{}$ radians.

This definition remains valid for any real angle, including negative angles and angles beyond one revolution.

Substitution into the circle equation gives the Pythagorean identity

$$\cos^2\theta+\sin^2\theta=1.$$

A common misconception is reversing the coordinates. Cosine is the $x$-coordinate; sine is the $y$-coordinate.

Exam technique

Identify the quadrant before choosing signs, then verify the Pythagorean identity. On HL papers, assume radians unless a degree symbol is shown; every Math AI paper is GDC-active.