Trigonometric identitiesAQA A-Level Maths: Subtopic test

1. 1
   The angle $\theta$ is obtuse and $\sin\theta=\frac{5}{13}$.

   (a)

   Find the value of $\cos\theta$.

   [1 mark]

   • A$\frac{12}{13}$
   • B$\frac{8}{13}$
   • C$-\frac{12}{13}$
   • D$-\frac{13}{12}$

   (b)

   Find the value of $\tan\theta$.

   [1 mark]

   • A$-\frac{5}{12}$
   • B$\frac{5}{12}$
   • C$-\frac{12}{5}$
   • D$-\frac{5}{13}$

   (c)

   Find the exact value of $\sin\theta\cos\theta$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   For all values of $x$ for which each expression is defined, let $A=\cos x\tan x$ and $B=\frac{\sin x}{\tan x}$.

   (a)

   Simplify $A$.

   [1 mark]

   • A$\cos x$
   • B$\sin x$
   • C$\frac{\cos^2x}{\sin x}$
   • D$1$

   (b)

   Simplify $B$.

   [1 mark]

   • A$\sin x$
   • B$\frac{\sin^2x}{\cos x}$
   • C$\frac{1}{\cos x}$
   • D$\cos x$

   (c)

   Show that $A^2+B^2=1$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The angle $\theta$ is acute and satisfies $\sin\theta=2\cos\theta$.

   (a)

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

   [3 marks]

   (b)

   Hence, or otherwise, find the exact value of $\dfrac{3\sin\theta-\cos\theta}{\sin\theta+4\cos\theta}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In this question, $\theta$ is an angle for which every expression is defined.

   (a)

   Prove that $\dfrac{\sin\theta}{1+\cos\theta}+\dfrac{1+\cos\theta}{\sin\theta}\equiv\dfrac{2}{\sin\theta}$.

   [6 marks]

   (b)

   (i) Show that $\tan\theta+\dfrac{1}{\tan\theta}\equiv\dfrac{1}{\sin\theta\cos\theta}$.
   (ii) Hence find the exact value of $\tan\theta+\dfrac{1}{\tan\theta}$ when $\theta$ is acute and $\sin\theta=\frac13$.

   [6 marks]

   Total for question 4: 12 marks

End of questions