Trigonometric identitiesEdexcel International A Level Maths: Subtopic test

1. 1
   $\theta$ is an obtuse angle with $\sin\theta=\frac{8}{17}$.

   (a)

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

   [1 mark]

   • A$-\frac{15}{17}$
   • B$\frac{15}{17}$
   • C$-\frac{225}{289}$
   • D$-\frac{8}{17}$

   (b)

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

   [1 mark]

   • A$\frac{8}{15}$
   • B$-\frac{8}{15}$
   • C$-\frac{15}{8}$
   • D$\frac{15}{8}$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   $\cos x=-\frac23$ and $180^\circ<x<270^\circ$.

   (a)

   Find the exact value of $\sin x$.

   [1 mark]

   • A$\frac{\sqrt5}{3}$
   • B$-\frac59$
   • C$-\frac{\sqrt5}{3}$
   • D$-\frac{\sqrt5}{2}$

   (b)

   Find the exact value of $\tan x$.

   [1 mark]

   • A$-\frac{\sqrt5}{2}$
   • B$\frac{2}{\sqrt5}$
   • C$\frac{\sqrt5}{3}$
   • D$\frac{\sqrt5}{2}$

   (c)

   Find the exact value of $\tan^2x-\sin^2x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   In this question, $\theta$ is any angle for which the expressions are defined.

   (a)

   Show that $(\sin\theta+\cos\theta)^2\equiv1+2\sin\theta\cos\theta$.

   [3 marks]

   (b)

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

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The two parts below are unrelated. $\theta$ and $x$ are angles for which every expression is defined.

   (a)

   Given that $4\sin^2\theta-\cos^2\theta=2$, show that $\tan^2\theta=\frac32$, and hence find the exact possible values of $\tan\theta$.

   [6 marks]

   (b)

   Prove that $\tan^2x-\sin^2x\equiv\tan^2x\sin^2x$.

   [6 marks]

   Total for question 4: 12 marks

End of questions