IEB Grade 11 Trigonometry Cheat Sheet

Trigonometry is about 50 marks of Grade 11 Paper 2. Every rule is on these two pages.

Reduction and co-functions

• CAST: All (Q1), Sin (Q2), Tan (Q3), Cos (Q4).

• $180°\pm \theta$, $360°\pm \theta$: same ratio, sign from CAST. Negative angles: $\sin (-\theta )=-\sin \theta$, $\cos (-\theta )=\cos \theta$.

• $90°\pm \theta$: sine and cosine swap;
  $\cos (90°+\theta )=-\sin \theta$.

• Given $\tan \theta$ and a quadrant: sketch $x$, $y$ with signs, find $r$ by Pythagoras.

Special angles

• $\sin 30°=\frac{1}{2}$, $\cos 30°=\frac{\sqrt{3}}{2}$, $\tan 30°=\frac{1}{\sqrt{3}}$

• $\sin 45°=\cos 45°=\frac{1}{\sqrt{2}}$, $\tan 45°=1$

• $\sin 60°=\frac{\sqrt{3}}{2}$, $\cos 60°=\frac{1}{2}$, $\tan 60°=\sqrt{3}$

Identities and equations

• $\tan \theta =\frac{\sin \theta }{\cos \theta }$;
  ${\sin }^2\theta +{\cos }^2\theta =1$

• Prove one side at a time; undefined where any denominator is 0.

• Find the reference angle $\theta$ from the calculator ($\theta ={\sin }^{-1}c$, ${\cos }^{-1}c$ or ${\tan }^{-1}c$). $\sin x=c$: $x=\theta +k\cdot 360°$ or $180°-\theta +k\cdot 360°$. $\cos x=c$: $x=\pm \theta +k\cdot 360°$. $\tan x=c$: $x=\theta +k\cdot 180°$. $k\in \mathbb{Z}$.

• Factorise; never divide by a ratio that could be 0.

Graphs

• $y=a\sin kx+q$ and $y=a\cos kx+q$: period $\frac{360°}{|k|}$, amplitude $|a|$, range $q-|a|\le y\le q+|a|$.

• $y=a\tan kx+q$: period $\frac{180°}{|k|}$, no amplitude, range all real numbers. ($k\ne 0$ throughout.)

• $\sin (x+p)$: shift $p$ left.

Triangles (2D only)

• Sine rule: a side and its opposite angle known; check the ambiguous case.

• Cosine rule: two sides and the included angle, or three sides.

• Area $=\frac{1}{2}ab\sin C$.

• Proofs of the sine, cosine and area rules use the height from one vertex.

Know by heart

• CAST and special angles

• The proofs of the sine, cosine and area rules

On the formula sheet

• Sine, cosine and area rules

Common traps

• Swapping sine and cosine for $180°\pm \theta$.

• Missing a family of solutions.

• Calculator in radians.

• Missing the obtuse case of the sine rule.