Edexcel C3 Specimen — Question 3 10 marks

Exam BoardEdexcel
ModuleC3 (Core Mathematics 3)
SessionSpecimen
Marks10
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TopicAddition & Double Angle Formulae
TypeEquation with half angles
DifficultyStandard +0.3 This is a structured multi-part question with clear scaffolding. Part (a) is a standard double angle formula derivation using A=B=θ/2. Part (b) requires algebraic manipulation using the result from (a) and standard identities, but follows a predictable pattern. Part (c) is straightforward once (b) is established—setting the factored form to zero and solving basic trigonometric equations. The question guides students through each step, making it slightly easier than average despite involving half-angle formulae.
Spec1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals1.05p Proof involving trig: functions and identities
3. (a) Using the identity for \(\cos ( A + B )\), prove that \(\cos \theta \equiv 1 - 2 \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right)\).
(b) Prove that \(1 + \sin \theta - \cos \theta \equiv 2 \sin \left( \frac { 1 } { 2 } \theta \right) \left[ \cos \left( \frac { 1 } { 2 } \theta \right) + \sin \left( \frac { 1 } { 2 } \theta \right) \right]\).
(c) Hence, or otherwise, solve the equation $$1 + \sin \theta - \cos \theta = 0 , \quad 0 \leq \theta < 2 \pi$$ 
3. (a) Using the identity for $\cos ( A + B )$, prove that $\cos \theta \equiv 1 - 2 \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right)$.\\
(b) Prove that $1 + \sin \theta - \cos \theta \equiv 2 \sin \left( \frac { 1 } { 2 } \theta \right) \left[ \cos \left( \frac { 1 } { 2 } \theta \right) + \sin \left( \frac { 1 } { 2 } \theta \right) \right]$.\\
(c) Hence, or otherwise, solve the equation

$$1 + \sin \theta - \cos \theta = 0 , \quad 0 \leq \theta < 2 \pi$$

\hfill \mbox{\textit{Edexcel C3  Q3 [10]}}
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