| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2014 |
| Session | June |
| Marks | 5 |
| Topic | Trig Proofs |
| Type | Prove trigonometric identity |
| Difficulty | Moderate -0.8 This is a straightforward application of standard double angle formula (sin 2θ = 2sin θ cos θ) and basic trig ratio (cot θ = cos θ/sin θ). Students first find sin θ using Pythagoras identity, then substitute into formulas. Requires only routine recall and simple arithmetic with fractions, making it easier than average. |
| Spec | 1.05a Sine, cosine, tangent: definitions for all arguments1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae |
$\sin\theta = \dfrac{\sqrt{7}}{4}$
M1 Attempt to find numerical value of $\sin\theta$ – from right angled triangle or identities. Must be correct triangle/identity
$\sin 2\theta = 2\sin\theta\cos\theta$
M1 Use $\sin 2\theta = 2\sin\theta\cos\theta$ with numerical values. M0 if using numerical value for $\theta$ not $\sin\theta$. M0M1 is possible (e.g. assuming 3, 4, 5 $\Delta$)
$= 2 \times \dfrac{\sqrt{7}}{4} \times \dfrac{3}{4} = \dfrac{3\sqrt{7}}{8}$
A1 Obtain correct surd aef (must be single fraction)
$\cot\theta = \dfrac{\cos\theta}{\sin\theta} = \dfrac{3/4}{\sqrt{7}/4} = \dfrac{3}{\sqrt{7}}$
M1 Attempt to find $\cot\theta$, using numerical values. M0 if using numerical value for $\theta$ not $\tan\theta$. Could follow first M0
A1 [5] Obtain correct surd aef (must be single fraction)6 Given that the angle $\theta$ is acute and $\cos \theta = \frac { 3 } { 4 }$ find, without using a calculator, the exact value of $\sin 2 \theta$ and of $\cot \theta$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2014 Q6 [5]}}