Trigonometric graphs and identitiesEdexcel A-Level Maths: Subtopic test

1. 1
   The curve $C$ has equation $y=\sin(x+30^\circ)$, where $x$ is in degrees and $0\le x\le360$.

   (a)

   Which single transformation maps the curve $y=\sin x$ onto $C$?

   [1 mark]

   • AA translation of $30^\circ$ to the right
   • BA stretch parallel to the $x$-axis, scale factor $30$
   • CA translation of $30$ units upwards
   • DA translation of $30^\circ$ to the left

   (b)

   Find the $y$-intercept of $C$.

   [1 mark]

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

   (c)

   State the coordinates of the maximum point of $C$ in the given interval.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $f$ is defined by $f(x)=\tan2x$, where $x$ is measured in degrees.

   (a)

   What is the period of $f$?

   [1 mark]

   • A$90^\circ$
   • B$180^\circ$
   • C$360^\circ$
   • D$45^\circ$

   (b)

   What is the smallest positive value of $x$ at which $f(x)$ is undefined?

   [1 mark]

   • A$90^\circ$
   • B$45^\circ$
   • C$22.5^\circ$
   • D$180^\circ$

   (c)

   Describe fully the single transformation that maps the curve $y=\tan x$ onto the curve $y=\tan2x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The angle $\theta$ is acute and $\sin\theta=\frac35$.

   (a)

   Find the exact values of $\cos\theta$ and $\tan\theta$.

   [3 marks]

   (b)

   Prove that $\frac{\sin\theta}{1+\cos\theta}+\frac{1+\cos\theta}{\sin\theta}\equiv\frac{2}{\sin\theta}$, and hence find the exact value of the expression on the left-hand side for this $\theta$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In this question $x$ and $\theta$ are measured in degrees.

   (a)

   (i) Prove that $\frac{\sin^2\theta}{1-\cos\theta}\equiv1+\cos\theta$.
   (ii) Prove that $\tan\theta+\frac{1}{\tan\theta}\equiv\frac{1}{\sin\theta\cos\theta}$.

   [6 marks]

   (b)

   The curve $C$ has equation $y=2+3\cos(x-40^\circ)$ for $0\le x\le360$.
   (i) Describe a sequence of one stretch and two translations that maps $y=\cos x$ onto $C$.
   (ii) State the maximum value of $y$ and the value of $x$ at which it occurs.
   (iii) State the minimum value of $y$.

   [6 marks]

   Total for question 4: 12 marks

End of questions