P2: TrigonometryEdexcel International A Level Maths: Topic test

1. 1
   $\tan\theta=-\dfrac{5}{12}$ and $90^\circ<\theta<180^\circ$.

   (a)

   Find the value of $\sin\theta$.

   [1 mark]

   • A$\frac{5}{13}$
   • B$-\frac5{13}$
   • C$\frac5{12}$
   • D$\frac{12}{13}$

   (b)

   Find the value of $\cos\theta$.

   [1 mark]

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

   (c)

   Find the exact value of $\dfrac{1}{\cos^2\theta}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   $\mathrm{f}(\theta)=\dfrac{\sin\theta}{\tan\theta}+\sin\theta\tan\theta$, for values of $\theta$ for which $\mathrm{f}(\theta)$ is defined.

   (a)

   Which expression is equal to $\dfrac{\sin\theta}{\tan\theta}$?

   [1 mark]

   • A$\tan\theta$
   • B$\dfrac{\sin^2\theta}{\cos\theta}$
   • C$\cos\theta$
   • D$\sin\theta\cos\theta$

   (b)

   Which expression is equal to $\mathrm{f}(\theta)$?

   [1 mark]

   • A$1$
   • B$\dfrac{1}{\sin\theta}$
   • C$\dfrac{2}{\cos\theta}$
   • D$\dfrac{1}{\cos\theta}$

   (c)

   Solve $\mathrm{f}(\theta)=2$ for $0^\circ\le\theta<360^\circ$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   $\mathrm{f}(x)=\sin\left(2x+30^\circ\right)$, for $0^\circ\le x<360^\circ$.

   (a)

   Solve $\mathrm{f}(x)=0.5$.

   [3 marks]

   (b)

   Solve $\mathrm{f}(x)=-0.6$, giving your answers to $1$ decimal place.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   $2\cos^2x-\sin x=1$, for $0^\circ\le x<360^\circ$.

   (a)

   Show that the equation can be written as $2\sin^2x+\sin x-1=0$, and hence solve it for $0^\circ\le x<360^\circ$.

   [6 marks]

   (b)

   Hence solve $2\cos^2(x-20^\circ)-\sin(x-20^\circ)=1$ for $0^\circ\le x<360^\circ$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   $\mathrm{g}(\theta)=\dfrac{\cos^2\theta}{1-\sin\theta}$, for $\sin\theta\ne1$.

   (a)

   Which expression is equal to $\mathrm{g}(\theta)$?

   [1 mark]

   • A$1-\sin\theta$
   • B$1+\sin\theta$
   • C$\cos\theta$
   • D$\dfrac{1}{1-\sin\theta}$

   (b)

   For $0^\circ\le\theta\le90^\circ$, which value of $\theta$ satisfies $\mathrm{g}(\theta)=1.5$?

   [1 mark]

   • A$30^\circ$
   • B$60^\circ$
   • C$48.6^\circ$
   • D$41.8^\circ$

   (c)

   Solve $\mathrm{g}(\theta)=0$ for $0^\circ\le\theta<360^\circ$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   $\mathrm{h}(x)=\cos\left(x+\dfrac{\pi}{6}\right)$, for $-\pi\le x\le\pi$, where $x$ is in radians.

   (a)

   Between which values must $x+\frac\pi6$ lie?

   [1 mark]

   • A$-\pi\le x+\frac\pi6\le\pi$
   • B$-\frac{7\pi}{6}\le x+\frac\pi6\le\frac{5\pi}{6}$
   • C$-\frac{\pi}{6}\le x+\frac\pi6\le\frac{7\pi}{6}$
   • D$-\frac{5\pi}{6}\le x+\frac\pi6\le\frac{7\pi}{6}$

   (b)

   Which gives all the solutions of $\mathrm{h}(x)=\frac12$?

   [1 mark]

   • A$x=\frac\pi6$ only
   • B$x=\pm\frac\pi3$
   • C$x=-\frac\pi2$ and $x=\frac\pi6$
   • D$x=-\frac\pi6$ and $x=\frac\pi2$

   (c)

   Solve $\mathrm{h}(x)=-\dfrac{\sqrt3}{2}$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   $\dfrac{\cos\theta}{1+\sin\theta}+\dfrac{1+\sin\theta}{\cos\theta}\equiv\dfrac{2}{\cos\theta}$.

   (a)

   Prove the identity.

   [3 marks]

   (b)

   Hence solve $\dfrac{\cos\theta}{1+\sin\theta}+\dfrac{1+\sin\theta}{\cos\theta}=5$ for $0^\circ\le\theta<360^\circ$, giving your answers to $1$ decimal place.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A passenger $P$ on a Ferris wheel has height $h=12-10\cos(30t)^\circ$ metres after $t$ minutes, for $0\le t\le12$. A second passenger $Q$ has height $k=12-7\sin(30t)^\circ$ metres.

   (a)

   Find the times at which $P$ is at a height of $17$ metres.

   [6 marks]

   (b)

   Show that $P$ and $Q$ are at the same height when $\tan(30t)^\circ=\frac{10}{7}$, and find the times in the first $12$ minutes when this happens, giving your answers to $2$ decimal places.

   [6 marks]

   Total for question 8: 12 marks

End of questions