| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2016 |
| Session | Specimen |
| Marks | 8 |
| Topic | Trig Proofs |
| Type | Solve equation using proven identity |
| Difficulty | Standard +0.3 Part (i) is a standard trigonometric identity proof requiring algebraic manipulation with common denominators. Part (ii) applies the proven identity with a compound angle substitution, then solves a straightforward equation for cosec. This is slightly easier than average as it's a guided two-part question where the first part directly enables the second, requiring only routine application of the identity and basic equation solving. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals1.05p Proof involving trig: functions and identities |
**Question 10(i)**
- Dealing with $\cot$ — B1
- Adding fractions in terms of $\sin$ and $\cos$ — M1
- Use of $\cos^2 + \sin^2$ — M1
- Simplification to given answer — A1
**Question 10(ii)**
- Substituting $\cosec\!\left(\theta + \dfrac{\pi}{4}\right)$ — M1
- Converting equation in $\sin$ — M1
- $\theta + \dfrac{\pi}{4} = 0.4115,\ 2.730,\ 6.695$ — M1
- $\theta = 1.94,\ 5.91$ — A1
**Total: 8 marks**10 (i) Prove that $\cot \theta + \frac { \sin \theta } { 1 + \cos \theta } = \operatorname { cosec } \theta$.\\
(ii) Hence solve the equation $\cot \left( \theta + \frac { \pi } { 4 } \right) + \frac { \sin \left( \theta + \frac { \pi } { 4 } \right) } { 1 + \cos \left( \theta + \frac { \pi } { 4 } \right) } = \frac { 5 } { 2 }$ for $0 \leqslant \theta \leqslant 2 \pi$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2016 Q10 [8]}}