IEB Grade 12 Trigonometry Cheat Sheet

Trigonometry is the biggest section of Paper 2 (about 50 marks). Everything on two pages.

Reduction and special angles

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

• $180°\pm \theta$ and $360°\pm \theta$: same ratio, sign from CAST.

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

• $\sin 30°=\cos 60°=\frac{1}{2}$; $\sin 60°=\cos 30°=\frac{\sqrt{3}}{2}$; $\sin 45°=\cos 45°=\frac{1}{\sqrt{2}}$; $\tan 45°=1$; $\tan 60°=\sqrt{3}$.

Identities

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

• $\sin (\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta$

• $\cos (\alpha \pm \beta )=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta$

• $\sin 2x=2\sin x\cos x$; $\cos 2x={\cos }^{2}x-{\sin }^{2}x=1-2{\sin }^{2}x=2{\cos }^{2}x-1$

• Proofs: one side at a time; state where undefined (any denominator $=0$).

Equations

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

• $\cos x=c$: $x=\pm \theta +k\cdot 360°$. $\tan x=c$: $x=\theta +k\cdot 180°$. Always '$k\in \mathbb{Z}$'.

• Never divide by $\sin x$ or $\cos x$; factorise.

• Multiple angles: divide the period too.

Graphs

• $y=a\sin k(x+p)+q$: amplitude $|a|$, period $\frac{360°}{|k|}$ ($\frac{180°}{|k|}$ for tan), shift $p$ left. ($k\ne 0$.)

Triangles

• Sine rule $\frac{a}{\sin A}=\frac{b}{\sin B}$: side and opposite angle known. Check the ambiguous case.

• Cosine rule $a^2=b^2+c^2-2bc\cos A$: two sides and included angle, or three sides.

• Area $=\frac{1}{2}ab\sin C$.

• 3D: find the side shared by the horizontal and vertical triangles first.

Know by heart

• CAST and special angles

• How to use the compound angle expansions for exact values (the derivation is not examined)

• Which form of $\cos 2x$ to use

On the formula sheet

• Compound and double angle expansions

• Sine, cosine and area rules

Common traps

• Swapping sin and cos for $180°\pm \theta$.

• Missing the second family of solutions.

• Calculator in radian mode.

• Using the cosine rule with a non-included angle.

Related topics

• reduction formulae
• compound angle identities
• double angle identities
• proving trig identities
• general solution trig equations
• trig graphs
• sine cosine and area rules
• 2d and 3d trigonometry