Compound and double angle formulaeAQA A-Level Maths: Subtopic test

1. 1
   Angles $A$ and $B$ are acute, with $\sin A=\frac35$ and $\cos B=\frac{5}{13}$.

   (a)

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

   [1 mark]

   • A$-\frac{16}{65}$
   • B$\frac{56}{65}$
   • C$\frac{16}{65}$
   • D$\frac{4}{13}$

   (b)

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

   [1 mark]

   • A$\frac{63}{65}$
   • B$\frac{33}{65}$
   • C$-\frac{33}{65}$
   • D$\frac{3}{13}$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The angle $\theta$ is acute and $\tan\theta=\frac34$.

   (a)

   Find the value of $\sin2\theta$.

   [1 mark]

   • A$\frac32$
   • B$\frac{24}{25}$
   • C$\frac{7}{25}$
   • D$\frac{12}{25}$

   (b)

   Find the value of $\cos2\theta$.

   [1 mark]

   • A$\frac{24}{25}$
   • B$-\frac{7}{25}$
   • C$\frac{32}{25}$
   • D$\frac{7}{25}$

   (c)

   Find the exact value of $\tan2\theta$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   In this question, work without a calculator and give exact answers. Note that $75^\circ=45^\circ+30^\circ$.

   (a)

   Show that $\sin75^\circ=\frac{\sqrt6+\sqrt2}{4}$.

   [3 marks]

   (b)

   Hence show that $\tan75^\circ=2+\sqrt3$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Use the double angle formulae. Give angles in radians.

   (a)

   Solve $\cos2\theta+3\sin\theta=2$ for $0\le\theta<2\pi$.

   [6 marks]

   (b)

   (i) Prove that $\frac{\sin2\theta}{1+\cos2\theta}\equiv\tan\theta$.
   (ii) Hence find the exact value of $\tan\frac{\pi}{8}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions