| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2009 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Prove algebraic trigonometric identity |
| Difficulty | Standard +0.8 Part (i) is a standard compound angle proof. Part (ii) requires substituting the triple angle result into the double angle formula, involving careful algebraic manipulation of polynomials. Part (iii) demands recognizing that the equation simplifies to a factorizable form and interpreting solutions geometrically. The multi-step nature, polynomial manipulation, and solution interpretation elevate this above routine exercises. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals1.05p Proof involving trig: functions and identities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State \(\cos 2\theta\cos\theta - \sin 2\theta\sin\theta\) | B1 | |
| Use at least one of \(\cos 2\theta = 2\cos^2\theta - 1\) and \(\sin 2\theta = 2\sin\theta\cos\theta\) | B1 | |
| Attempt to express in terms of \(\cos\theta\) only | M1 | using correct identities for \(\cos 2\theta\), \(\sin 2\theta\) and \(\sin^2\theta\) |
| Obtain \(4\cos^3\theta - 3\cos\theta\) | A1 | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply \(\cos 6\theta = 2\cos^2 3\theta - 1\) | B1 | |
| Use expression for \(\cos 3\theta\) and attempt expansion | M1 | for expression of form \(\pm 2\cos^2 3\theta \pm 1\) |
| Obtain \(32c^6 - 48c^4 + 18c^2 - 1\) | A1 | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State \(\cos 6\theta = 4\cos^3 2\theta - 3\cos 2\theta\) | B1 | maybe implied |
| Express \(\cos 2\theta\) in terms of \(\cos\theta\) and attempt expansion | M1 | for expression of form \(\pm 2\cos^2\theta \pm 1\) |
| Obtain \(32c^6 - 48c^4 + 18c^2 - 1\) | A1 | (3) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Substitute for \(\cos 6\theta\) | *M1 | with simplification attempted |
| Obtain \(32c^6 - 48c^4 = 0\) | A1 | or equiv |
| Attempt solution for \(c\) of equation | M1 | dep *M |
| Obtain \(c^2 = \frac{3}{2}\) and observe no solutions | A1 | or equiv; correct work only |
| Obtain \(c = 0\), give at least three specific angles and conclude odd multiples of 90 | A1 | 5 |
# Question 9:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State $\cos 2\theta\cos\theta - \sin 2\theta\sin\theta$ | B1 | |
| Use at least one of $\cos 2\theta = 2\cos^2\theta - 1$ and $\sin 2\theta = 2\sin\theta\cos\theta$ | B1 | |
| Attempt to express in terms of $\cos\theta$ only | M1 | using correct identities for $\cos 2\theta$, $\sin 2\theta$ and $\sin^2\theta$ |
| Obtain $4\cos^3\theta - 3\cos\theta$ | A1 | **4** | AG; necessary detail required |
## Part (ii):
**Either method:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply $\cos 6\theta = 2\cos^2 3\theta - 1$ | B1 | |
| Use expression for $\cos 3\theta$ and attempt expansion | M1 | for expression of form $\pm 2\cos^2 3\theta \pm 1$ |
| Obtain $32c^6 - 48c^4 + 18c^2 - 1$ | A1 | **3** | AG; necessary detail required |
**Or:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| State $\cos 6\theta = 4\cos^3 2\theta - 3\cos 2\theta$ | B1 | maybe implied |
| Express $\cos 2\theta$ in terms of $\cos\theta$ and attempt expansion | M1 | for expression of form $\pm 2\cos^2\theta \pm 1$ |
| Obtain $32c^6 - 48c^4 + 18c^2 - 1$ | A1 | **(3)** | AG; necessary detail required |
## Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Substitute for $\cos 6\theta$ | *M1 | with simplification attempted |
| Obtain $32c^6 - 48c^4 = 0$ | A1 | or equiv |
| Attempt solution for $c$ of equation | M1 | dep *M |
| Obtain $c^2 = \frac{3}{2}$ and observe no solutions | A1 | or equiv; correct work only |
| Obtain $c = 0$, give at least three specific angles and conclude odd multiples of 90 | A1 | **5** | AG; or equiv; necessary detail required; correct work only |9 (i) By first expanding $\cos ( 2 \theta + \theta )$, prove that
$$\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta$$
(ii) Hence prove that
$$\cos 6 \theta \equiv 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$$
(iii) Show that the only solutions of the equation
$$1 + \cos 6 \theta = 18 \cos ^ { 2 } \theta$$
are odd multiples of $90 ^ { \circ }$.
\hfill \mbox{\textit{OCR C3 2009 Q9 [12]}}