The t-formulaeEdexcel A-Level Further Maths: Subtopic test

1. 1
   Given that $\theta$ is acute and $\tan\frac{\theta}{2}=\frac12$.

   (a)

   Find the value of $\sin\theta$.

   [1 mark]

   • A$\frac35$
   • B$\frac45$
   • C$1$
   • D$\frac43$

   (b)

   Find the value of $\tan\theta$.

   [1 mark]

   • A$\frac34$
   • B$\frac45$
   • C$\frac43$
   • D$1$

   (c)

   Find the exact value of $\operatorname{cosec}\theta+\cot\theta$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $t=\tan\frac{x}{2}$, where $0<x<\pi$ and $x\neq\frac{\pi}{2}$.

   (a)

   Which expression is equal to $1+\cos x$ in terms of $t$?

   [1 mark]

   • A$\frac{2t^2}{1+t^2}$
   • B$\frac{2t}{1+t^2}$
   • C$\frac{2}{1+t^2}$
   • D$\frac{2}{1-t^2}$

   (b)

   Which expression is equal to $\dfrac{1-\cos x}{\sin x}$ in terms of $t$?

   [1 mark]

   • A$t^2$
   • B$\frac1t$
   • C$2t$
   • D$t$

   (c)

   Show that $\sec x+\tan x=\dfrac{1+t}{1-t}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the equation $3\cos x+4\sin x=2$ for $0\le x<2\pi$, and let $t=\tan\frac{x}{2}$.

   (a)

   Show that the equation can be written as $5t^2-8t-1=0$.

   [3 marks]

   (b)

   Hence solve the equation, giving your values of $x$ in radians to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $t=\tan\frac{\theta}{2}$, where $0<\theta<\pi$ and $\theta\neq\frac{\pi}{2}$.

   (a)

   Show that $\dfrac{1+\operatorname{cosec}\theta}{\cot\theta}\equiv\dfrac{1+t}{1-t}$.

   [6 marks]

   (b)

   (i) Use the identity in part (a) to solve $\dfrac{1+\operatorname{cosec}\theta}{\cot\theta}=3$, giving $\theta$ in radians to 3 significant figures.
   (ii) Find the exact value of $\dfrac{1+\operatorname{cosec}\theta}{\cot\theta}$ when $\theta=\dfrac{\pi}{3}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions