Pure: TrigonometryEdexcel A-Level Maths: Topic test

1. 1
   In triangle $PQR$, $PQ=8$ cm, $PR=11$ cm and angle $QPR=50^\circ$.

   (a)

   What is the length of $QR$, to 3 significant figures?

   [1 mark]

   • A$7.00$ cm
   • B$13.6$ cm
   • C$8.48$ cm
   • D$71.9$ cm

   (b)

   What is the area of triangle $PQR$, to 3 significant figures?

   [1 mark]

   • A$28.3$ cm$^2$
   • B$33.7$ cm$^2$
   • C$67.4$ cm$^2$
   • D$44.0$ cm$^2$

   (c)

   Find angle $PQR$, giving your answer to the nearest $0.1^\circ$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A flower bed is a sector of a circle of radius $9$ m. The angle at the centre of the sector is $1.4$ radians.

   (a)

   What is the length of the curved edge of the bed?

   [1 mark]

   • A$12.6$ m
   • B$56.7$ m
   • C$6.43$ m
   • D$25.2$ m

   (b)

   What is the area of the bed?

   [1 mark]

   • A$12.6$ m$^2$
   • B$113.4$ m$^2$
   • C$6.3$ m$^2$
   • D$56.7$ m$^2$

   (c)

   The bed is redesigned with the same radius so that its area is exactly $81$ m$^2$. Find the new angle at the centre, in radians.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=2\sin3x+1$, where $x$ is measured in degrees.

   (a)

   Write down the amplitude of the curve, the period of the curve and the maximum value of $y$.

   [3 marks]

   (b)

   Solve $2\sin3x+1=2$ for $0\le x\le180^\circ$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $g(x)=8\cos x+15\sin x$, where $x$ is measured in degrees.

   (a)

   Express $g(x)$ in the form $R\cos(x-\alpha)$, where $R>0$ and $0<\alpha<90^\circ$, giving $\alpha$ to 1 decimal place. Hence state the maximum value of $g(x)$, the smallest positive value of $x$ at which it occurs, and the smallest positive value of $x$ at which $g(x)$ is a minimum.

   [6 marks]

   (b)

   Hence solve $8\cos x+15\sin x=10$ for $0\le x<360^\circ$, giving your answers to 1 decimal place.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The functions $f$ and $g$ are defined by $f(x)=\arcsin x$ for $-1\le x\le1$ and $g(x)=\mathrm{cosec}\,x$ for $0<x<\pi$, where $x$ is in radians.

   (a)

   What is the range of $f$?

   [1 mark]

   • A$0\le f(x)\le\pi$
   • B$-1\le f(x)\le1$
   • C$-\pi\le f(x)\le\pi$
   • D$-\frac{\pi}{2}\le f(x)\le\frac{\pi}{2}$

   (b)

   What is the range of $g$?

   [1 mark]

   • A$g(x)\ge1$
   • B$-1\le g(x)\le1$
   • C$g(x)>0$
   • D$g(x)\le-1$ or $g(x)\ge1$

   (c)

   Find the exact value of $f\left(-\frac{\sqrt3}{2}\right)+g\left(\frac{5\pi}{6}\right)$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A surveyor stands $900$ m from the foot of a tower of height $40$ m on level ground. The angle of elevation of the top of the tower is $\theta$ radians, which is small.

   (a)

   Using a small angle approximation, which is the best estimate of $\theta$?

   [1 mark]

   • A$0.0444$
   • B$0.0222$
   • C$22.5$
   • D$2.55$

   (b)

   Using $\cos\theta\approx1-\frac{\theta^2}{2}$ with $\theta=\frac{40}{900}$, which is the approximate value of $1-\cos\theta$?

   [1 mark]

   • A$0.0444$
   • B$9.88\times10^{-4}$
   • C$1.98\times10^{-3}$
   • D$0.999$

   (c)

   Use the small angle approximations to show that, for small $\theta$, $\dfrac{6\theta\sin\theta}{1-\cos3\theta}\approx\dfrac43$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The voltage $V$ volts in an alternating supply is modelled by $V=325\sin(100\pi t)$, where $t$ is the time in seconds and the angle is in radians.

   (a)

   Find the period of $V$ and the number of complete cycles in one second. Find also the first positive time at which $V$ reaches its maximum value.

   [3 marks]

   (b)

   Find the first two positive times at which $V=200$, giving your answers to 3 significant figures.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Angles in this question are measured in degrees and $0\le x<360^\circ$.

   (a)

   Solve $2\sec^2x+\tan x=5$, giving your answers to 1 decimal place where necessary.

   [6 marks]

   (b)

   (i) By writing $3x$ as $2x+x$, prove that $\cos3x\equiv4\cos^3x-3\cos x$. (ii) Hence find the exact value of $\cos3x$ when $\cos x=\frac34$.

   [6 marks]

   Total for question 8: 12 marks

End of questions