Trigonometry: past paper questions

Where the questions come from

• Question 3Oct/Nov 2025, paper 31Structured8 marksHas a diagram

  Express $3\sqrt{2}\sin (x+{45}^{\circ })+\cos x$ in the form $R\cos (x-\alpha )$, where $R>0$ and $0^{\circ }<\alpha <{90}^{\circ }$.

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• Question 5Oct/Nov 2023, paper 31Structured6 marks

  Given that $$\sin (x+\frac{1}{6}\pi )-\sin (x-\frac{1}{6}\pi )=\cos (x+\frac{1}{3}\pi )-\cos (x-\frac{1}{3}\pi ),$$ find the exact value of $\tan x$.

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• Question 7Oct/Nov 2024, paper 32Structured6 marks

  Show that the equation ${\tan }^{3}x+2\tan 2x-\tan x=0$ may be expressed as $${\tan }^{4}x-2{\tan }^{2}x-3=0$$ for $\tan x\mathrm{\ne }0$.

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• Question 7May/June 2025, paper 31Structured8 marks

  Express $5\sin (x+\frac{1}{6}\pi )-4\cos x$ in the form $R\sin (x-\alpha )$, where $R>0$ and $0<\alpha <\frac{1}{2}\pi$.

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• Question 4May/June 2023, paper 31Structured6 marks

  Show that the equation $\sin 2\theta +\cos 2\theta =2{\sin }^2\theta$ can be expressed in the form $${\cos }^2\theta +2\sin \theta \cos \theta -3{\sin }^2\theta =0.$$

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• Question 7May/June 2025, paper 32Structured7 marks

  Express $7\sin \theta +24\cos \theta$ in the form $R\cos (\theta -\alpha )$, where $R>0$ and $0<\alpha <\frac{1}{2}\pi$.

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• Question 6May/June 2023, paper 33Structured8 marks

  Express $3\cos x+2\cos (x-{60}^{\circ })$ in the form $R\cos (x-\alpha )$, where $R>0$ and $0^{\circ }<\alpha <{90}^{\circ }$.

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• Question 7May/June 2024, paper 33Structured6 marks

  Let $\mathrm{f}(x)=8x^3+54x^2-17x-21$. Show that $x+7$ is a factor of $\mathrm{f}(x)$.

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• Question 8May/June 2024, paper 33Structured8 marksHas a diagram

  Express $3\cos 2x-\sqrt{3}\sin 2x$ in the form $R\cos (2x+\alpha )$, where $R>0$ and $0<\alpha <\frac{1}{2}\pi$.

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