The point reached by an anticlockwise angle theta on the unit circle is (costheta,sintheta). Clockwise angles are negative. Hence tantheta= sinthetacostheta, where costheta 0. **Quadrant** | **Positive ratios** I | Sine, cosine and tangent II | Sine III | Tangent IV | Cosine sin(pi-theta)=sintheta, (pi-theta)=-costheta, (pi-theta)=-tantheta sin(-theta)=-sintheta, (-theta)=costheta, (-theta)=-tantheta A nonvertical line through the origin making angle theta with the positive x-axis has gradient tantheta and equation y=xtantheta. The angle is not itself the gradient.

The point reached by an anticlockwise angle \(\theta\) on the unit circle is \((\cos\theta,\sin\theta)\). Clockwise angles are negative. Hence \(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\), where \(\cos\theta\ne0\).

\(\sin(\pi-\theta)=\sin\theta,\quad\cos(\pi-\theta)=-\cos\theta,\quad\tan(\pi-\theta)=-\tan\theta\)

\(\sin(-\theta)=-\sin\theta,\quad\cos(-\theta)=\cos\theta,\quad\tan(-\theta)=-\tan\theta\)

A nonvertical line through the origin making angle \(\theta\) with the positive \(x\)-axis has gradient \(\tan\theta\) and equation \(y=x\tan\theta\). The angle is not itself the gradient.

Trigonometry

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

The Unit Circle and Related Angles

The Unit Circle and Related Angles