Trigonometric functions and their graphsEdexcel International A Level Maths: Subtopic test

1. 1
   The function $f$ is defined by $f(x)=3\sin x$, where $x$ is in radians.

   (a)

   Which of the following gives the range of $f$?

   [1 mark]

   • A$-1\le f(x)\le1$
   • B$0\le f(x)\le3$
   • C$-3\le f(x)\le3$
   • D$-1\le f(x)\le3$

   (b)

   Find the smallest positive value of $x$ for which $f(x)=3$.

   [1 mark]

   • A$\pi$
   • B$\frac{3\pi}{2}$
   • C$3$
   • D$\frac{\pi}{2}$

   (c)

   Describe the single transformation that maps the curve $y=\sin x$ onto the curve $y=f(x)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $g$ is defined by $g(x)=\sin\left(x+\frac{\pi}{6}\right)$, where $x$ is in radians.

   (a)

   Find the value of $g(0)$.

   [1 mark]

   • A$\frac12$
   • B$\frac{\sqrt3}{2}$
   • C$1$
   • D$0$

   (b)

   Which transformation maps the curve $y=\sin x$ onto the curve $y=g(x)$?

   [1 mark]

   • Atranslation of $\frac{\pi}{6}$ to the right
   • Btranslation of $\frac{\pi}{6}$ to the left
   • Ctranslation of $\frac{\pi}{6}$ upwards
   • Dtranslation of $\frac{\pi}{6}$ downwards

   (c)

   Solve $g(x)=1$ for $0\le x\le2\pi$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $h$ is defined by $h(x)=\sin2x$, where $x$ is in radians.

   (a)

   State the period of $h$ and find all solutions of $h(x)=0$ in the interval $0\le x\le\pi$.

   [3 marks]

   (b)

   Solve $h(x)=\frac12$ for $0\le x\le2\pi$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=2\sin\left(x-\frac{\pi}{4}\right)$, where $x$ is in radians.

   (a)

   (i) Describe the sequence of two transformations that maps $y=\sin x$ onto $C$.
   (ii) State the maximum value of $y$ on $C$ and the smallest positive value of $x$ at which it occurs.
   (iii) Find the exact $y$-coordinate of the point where $C$ crosses the $y$-axis.

   [6 marks]

   (b)

   (i) Solve $2\sin\left(x-\frac{\pi}{4}\right)=\sqrt2$ for $0\le x\le2\pi$.
   (ii) Explain why there are no further solutions in this interval.

   [6 marks]

   Total for question 4: 12 marks

End of questions