P3: TrigonometryEdexcel International A Level Maths: Topic test

1. 1
   The angle $\theta$ is obtuse and $\cot\theta=-\dfrac{5}{12}$.

   (a)

   Find the value of $\mathrm{cosec}\,\theta$.

   [1 mark]

   • A$\dfrac{12}{13}$
   • B$\dfrac{13}{12}$
   • C$-\dfrac{13}{12}$
   • D$\dfrac{13}{5}$

   (b)

   Find the value of $\sec\theta$.

   [1 mark]

   • A$\dfrac{13}{5}$
   • B$-\dfrac{5}{13}$
   • C$\dfrac{13}{12}$
   • D$-\dfrac{13}{5}$

   (c)

   Find the exact value of $\sec^2\theta+\mathrm{cosec}^2\theta$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=\mathrm{cosec}\,x$ for $-\pi<x<\pi$, $x\neq0$, where $x$ is in radians.

   (a)

   Which of the following is the range of $\mathrm{f}$?

   [1 mark]

   • A$\mathrm{f}(x)\geqslant1$
   • B$-1\leqslant\mathrm{f}(x)\leqslant1$
   • C$\mathrm{f}(x)\leqslant-1$ or $\mathrm{f}(x)\geqslant1$
   • D$\mathrm{f}(x)\in\mathbb{R}$, $\mathrm{f}(x)\neq0$

   (b)

   How many solutions does the equation $\mathrm{f}(x)=0.5$ have?

   [1 mark]

   • A$0$
   • B$1$
   • C$2$
   • D$4$

   (c)

   Solve $\mathrm{f}(x)=2$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=\arctan(2x)$, $x\in\mathbb{R}$, where angles are in radians.

   (a)

   State the range of $\mathrm{f}$ and find $\mathrm{f}^{-1}(x)$.

   [3 marks]

   (b)

   Solve $\arctan(2x)+\arctan(x)=\dfrac\pi4$, giving $x$ in exact form and justifying any solution you reject.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $\mathrm{f}(\theta)=2\sec^2\theta-\tan\theta-5$, where $\theta$ is measured in degrees.

   (a)

   Solve $\mathrm{f}(\theta)=0$ for $0\leqslant\theta<360^{\circ}$, giving your answers to one decimal place.

   [6 marks]

   (b)

   Show that $\mathrm{cosec}\,2\theta+\cot2\theta\equiv\cot\theta$. Hence solve $\mathrm{cosec}\,2\theta+\cot2\theta=\dfrac23$ for $0<\theta<360^{\circ}$, giving your answers to one decimal place.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The angles $A$ and $B$ are acute, with $\sin A=\dfrac{8}{17}$ and $\cos B=\dfrac45$.

   (a)

   Find the value of $\sin(A+B)$.

   [1 mark]

   • A$\dfrac{77}{85}$
   • B$-\dfrac{13}{85}$
   • C$\dfrac{84}{85}$
   • D$\dfrac{32}{85}$

   (b)

   Find the value of $\cos2A$.

   [1 mark]

   • A$\dfrac{225}{289}$
   • B$\dfrac{240}{289}$
   • C$\dfrac{1}{17}$
   • D$\dfrac{161}{289}$

   (c)

   Find the exact value of $\tan(A-B)$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The expression $5\cos\theta-12\sin\theta$ is written in the form $R\cos(\theta+\alpha)$, where $R>0$ and $0<\alpha<90^{\circ}$.

   (a)

   Find the value of $R$.

   [1 mark]

   • A$7$
   • B$17$
   • C$13$
   • D$169$

   (b)

   Find the value of $\alpha$, to $1$ decimal place.

   [1 mark]

   • A$22.6^{\circ}$
   • B$67.4^{\circ}$
   • C$112.6^{\circ}$
   • D$-67.4^{\circ}$

   (c)

   Write down the minimum value of $5\cos\theta-12\sin\theta$ and the smallest positive value of $\theta$ at which it occurs.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A signal voltage $V$ volts at time $t$ seconds is modelled by $V=7\sin t-24\cos t$, where $t$ is measured in radians.

   (a)

   Express $V$ in the form $R\sin(t-\alpha)$, where $R>0$ and $0<\alpha<\dfrac\pi2$, giving $\alpha$ to $3$ decimal places.

   [3 marks]

   (b)

   Solve $V=10$ for $0\leqslant t<2\pi$, giving your answers to $2$ decimal places.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The function $\mathrm{g}$ is defined by $\mathrm{g}(\theta)=\sqrt3\sin2\theta-\cos2\theta$, $0\leqslant\theta\leqslant\pi$, where $\theta$ is in radians.

   (a)

   Express $\mathrm{g}(\theta)$ in the form $R\sin(2\theta-\alpha)$, where $R>0$ and $0<\alpha<\dfrac\pi2$. Hence solve $\mathrm{g}(\theta)=1$, giving your answers as exact multiples of $\pi$.

   [6 marks]

   (b)

   Use double angle formulae to show that the equation $\mathrm{g}(\theta)=1$ can be written as $\cos\theta\left(\sqrt3\sin\theta-\cos\theta\right)=0$. Hence solve the equation, and confirm that your solutions agree with part (a).

   [6 marks]

   Total for question 8: 12 marks

End of questions