Addition and double angle formulaeEdexcel International A Level Maths: Subtopic test

1. 1
   The angle $A$ is acute and $\sin A=\frac{3}{5}$.

   (a)

   Find the value of $\sin2A$.

   [1 mark]

   • A$\frac{24}{25}$
   • B$\frac65$
   • C$\frac{12}{25}$
   • D$\frac{7}{25}$

   (b)

   Find the value of $\cos2A$.

   [1 mark]

   • A$\frac{24}{25}$
   • B$-\frac{7}{25}$
   • C$\frac{7}{25}$
   • D$\frac15$

   (c)

   Find the exact value of $\tan2A$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The angle $75^{\circ}$ can be written as $45^{\circ}+30^{\circ}$, and exact values of the sine and cosine of $30^{\circ}$ and $45^{\circ}$ are known.

   (a)

   Find the exact value of $\sin75^{\circ}$.

   [1 mark]

   • A$\frac{\sqrt6-\sqrt2}{4}$
   • B$\frac{\sqrt6+\sqrt2}{4}$
   • C$\frac{\sqrt6+\sqrt2}{2}$
   • D$\frac{1+\sqrt3}{2}$

   (b)

   Find the exact value of $\cos75^{\circ}$.

   [1 mark]

   • A$\frac{\sqrt6+\sqrt2}{4}$
   • B$\frac{\sqrt6-\sqrt2}{2}$
   • C$\frac{\sqrt2-\sqrt6}{4}$
   • D$\frac{\sqrt6-\sqrt2}{4}$

   (c)

   Hence find $\tan75^{\circ}$ in the form $a+\sqrt b$, where $a$ and $b$ are integers.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Angles are measured in degrees.

   (a)

   Prove that $\cos x\cos2x+\sin x\sin2x\equiv\cos x$.

   [3 marks]

   (b)

   Solve $\sin2x=\sin x$ for $0\le x<360^{\circ}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A calculator may be used.

   (a)

   (i) Show that $\cos3x\equiv4\cos^3x-3\cos x$.
   (ii) Given that $\cos x=\frac34$, find the exact value of $\cos3x$.

   [6 marks]

   (b)

   The angle $\theta$ is acute and $\cos\theta=\frac{7}{25}$.
   (i) Use a double angle formula to find the exact value of $\cos\frac{\theta}{2}$.
   (ii) Hence find the exact value of $\tan\frac{\theta}{2}$.
   (iii) Use the double angle formula for tangent to find $\tan\theta$, and check that it agrees with $\frac{\sin\theta}{\cos\theta}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions