Trigonometry: past paper questions

Where the questions come from

• Question 2Oct/Nov 2025, paper 11Structured5 marks

  A geometric progression has first term $a$ and common ratio $\cos \theta$, where $0<\theta <\frac{1}{2}\pi$. It is given that the second term is 8 and the fifth term is $\frac{1}{8}$. Find the value of $\theta$.

  Try this question
• Question 8Oct/Nov 2024, paper 11Structured8 marks

  It is given that $\beta$ is an angle between ${90}^{\circ }$ and ${180}^{\circ }$ such that $\sin \beta =a$. Express ${\tan }^2\beta -3\sin \beta \cos \beta$ in terms of $a$.

  Try this question
• Question 5Oct/Nov 2023, paper 11Structured6 marks

  Show that the equation $$4\sin x+\frac{5}{\tan x}+\frac{2}{\sin x}=0$$ may be expressed in the form $a{\cos }^{2}x+b\cos x+c=0$, where $a$, $b$ and $c$ are integers to be found.

  Try this question
• Question 2Oct/Nov 2023, paper 12Structured2 marks

  Find the exact solution of the equation $$\frac{1}{6}\pi +{\tan }^{-1}(4x)=-{\cos }^{-1}\left(\frac{1}{2}\sqrt{3}\right)\mathrm{.}$$ Find the exact solution of the equation $$\frac{1}{6}\pi +{\tan }^{-1}(4x)=-{\cos }^{-1}\left(\frac{1}{2}\sqrt{3}\right)\mathrm{.}$$

  Try this question
• Question 2Oct/Nov 2024, paper 13Structured2 marks

  Find the exact solution of the equation cos 1/6π + tan 2x + (√3)/2 = 0 for -1/4π < x < 1/4π.

  Try this question
• Question 5Oct/Nov 2023, paper 12Structured6 marks

  The first, second and third terms of a geometric progression are $\sin \theta$, $\cos \theta$ and $2-\sin \theta$ respectively, where $\theta$ radians is an acute angle. Find the value of $\theta$.

  Try this question
• Question 2Oct/Nov 2025, paper 13Structured6 marks

  Solve the equation ${\tan }^{-1}(5x-3)=-\frac{1}{4}\pi$.

  Try this question
• Question 1May/June 2025, paper 11Structured4 marks

  Solve the equation $6\sin \theta =1+\frac{2}{\sin \theta }$ for $-{180}^{\circ }<\theta <{180}^{\circ }$.

  Try this question
• Question 3Oct/Nov 2023, paper 13Structured7 marks

  Show that the equation $$5\cos \theta -\sin \theta \tan \theta +1=0$$ may be expressed in the form $a{\cos }^2\theta +b\cos \theta +c=0$, where $a$, $b$ and $c$ are constants to be found.

  Try this question
• Question 3May/June 2024, paper 12Structured6 marks

  Show that the equation $\frac{7\tan \theta }{\cos \theta }+12=0$ can be expressed as $$12{\sin }^2\theta -7\sin \theta -12=0.$$

  Try this question
• Question 8Oct/Nov 2023, paper 13Structured5 marksHas a diagram

  The diagram shows part of the graph of $y=\sin (a(x+b))$, where $a$ and $b$ are positive constants. State the value of $a$ and one possible value of $b$.

  Try this question
• Question 5May/June 2024, paper 12Structured9 marks

  The first and second terms of an arithmetic progression are $\tan \theta$ and $\sin \theta$ respectively, where $0<\theta <\frac{1}{2}\pi$. Given that $\theta =\frac{1}{4}\pi$, find the exact sum of the first 40 terms of the progression.

  Try this question
• Question 1May/June 2023, paper 11Structured3 marks

  Solve the equation $4\sin \theta +\tan \theta =0$ for $0^{\circ }<\theta <{180}^{\circ }$.

  Try this question
• Question 2May/June 2025, paper 13Structured4 marks

  The first two terms of a geometric progression are $$4{\sin }^2\theta ,8{\sin }^3\theta ,$$ where $\theta$ is an angle such that $0<\theta <\frac{1}{6}\pi$. Given that the sum to infinity of the progression is $\frac{1}{2}$, find the value of $\theta$.

  Try this question
• Question 2May/June 2024, paper 13Structured5 marksHas a diagram

  The diagram shows the curve $y=k\cos (x-\frac{1}{6}\pi )$ where $k$ is a positive constant and $x$ is measured in radians. The curve crosses the $x$-axis at point $A$ and $B$ is a minimum point. Find the coordinates of $A$ and $B$.

  Try this question
• Question 5May/June 2025, paper 13Structured6 marks

  Solve the equation $$4\sin \theta \tan \theta =1+5\cos \theta$$ for $-{180}^{\circ }<\theta <{180}^{\circ }$.

  Try this question
• Question 4May/June 2024, paper 13Structured6 marks

  Show that the equation $\cos \theta (7\tan \theta -5\cos \theta )=1$ can be written in the form $a{\sin }^2\theta +b\sin \theta +c=0$, where $a$, $b$ and $c$ are integers to be found.

  Try this question
• Question 4May/June 2023, paper 13Structured7 marks

  Show that the equation $$3{\tan }^{2}x-3{\sin }^{2}x-4=0$$ may be expressed in the form $a{\cos }^{4}x+b{\cos }^{2}x+c=0$, where $a$, $b$ and $c$ are constants to be found.

  Try this question