| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Prove algebraic trigonometric identity |
| Difficulty | Moderate -0.8 Both parts are straightforward identity proofs requiring standard techniques. Part (i) uses the double angle formula for sin 2θ and simplification with tan θ = sin θ/cos θ, leading directly to a Pythagorean identity. Part (ii) applies compound angle formulas with the special angle 45° where sin 45° = cos 45° = 1/√2, making the algebra trivial. These are textbook exercises testing recall and basic manipulation rather than problem-solving or insight. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.05p Proof involving trig: functions and identities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| LHS \(= \frac{2\sin\theta\cos\theta}{2\sin\theta/\cos\theta} + \sin^2\theta\) | M1 | |
| \(= \cos^2\theta + \sin^2\theta\) | A1 | |
| \(= 1 =\) RHS | E1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\sin(x+45°) = \frac{1}{\sqrt{2}}\sin x + \frac{1}{\sqrt{2}}\cos x\) | M1, A1 | |
| \(= \cos(x - 45°)\) | B1 | |
| or \(\sin(x+45°) = \cos(90°-[x+45°]) = \cos(45°-x) = \cos(x-45°)\) (even function) | M1, A1, E1 | |
| Total: 3 |
## Question 6(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| LHS $= \frac{2\sin\theta\cos\theta}{2\sin\theta/\cos\theta} + \sin^2\theta$ | M1 | |
| $= \cos^2\theta + \sin^2\theta$ | A1 | |
| $= 1 =$ RHS | E1 | |
| **Total: 3** | | |
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## Question 6(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\sin(x+45°) = \frac{1}{\sqrt{2}}\sin x + \frac{1}{\sqrt{2}}\cos x$ | M1, A1 | |
| $= \cos(x - 45°)$ | B1 | |
| **or** $\sin(x+45°) = \cos(90°-[x+45°]) = \cos(45°-x) = \cos(x-45°)$ (even function) | M1, A1, E1 | |
| **Total: 3** | | |
---6 Prove that\\
(i) $\frac { \sin 2 \theta } { 2 \tan \theta } + \sin ^ { 2 } \theta = 1$,\\
(ii) $\quad \sin \left( x + 45 ^ { \circ } \right) = \cos \left( x - 45 ^ { \circ } \right)$.
\hfill \mbox{\textit{OCR MEI C4 Q6 [6]}}