Trigonometric proofs and applicationsAQA A-Level Maths: Subtopic test

1. 1
   A student sets out to prove the identity $\frac{\sin2\theta}{1+\cos2\theta}\equiv\tan\theta$.

   (a)

   Which expression is equal to $1+\cos2\theta$?

   [1 mark]

   • A$2\cos^2\theta$
   • B$2\sin^2\theta$
   • C$2\cos^2\theta-2$
   • D$2\cos\theta$

   (b)

   After substituting the double angle formulae the left-hand side is $\frac{2\sin\theta\cos\theta}{2\cos^2\theta}$. Which expression does this simplify to, completing the proof?

   [1 mark]

   • A$\cot\theta$
   • B$\tan\theta$
   • C$\sin\theta\cos\theta$
   • D$\tan2\theta$

   (c)

   State the values of $\theta$ in the interval $0\le\theta\le2\pi$ for which the identity is not valid because both sides are undefined.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $f(x)=\sin^4x-\cos^4x$, where $x$ is in radians.

   (a)

   Which is a correct factorisation of $\sin^4x-\cos^4x$?

   [1 mark]

   • A$(\sin^2x-\cos^2x)^2$
   • B$(\sin x-\cos x)^4$
   • C$(\sin^2x-\cos^2x)(\sin^2x+\cos^2x)$
   • D$(\sin^2x+\cos^2x)^2$

   (b)

   Which of these is equal to $f(x)$ for all $x$?

   [1 mark]

   • A$\cos2x$
   • B$-\cos2x$
   • C$1$
   • D$\sin2x$

   (c)

   Hence solve $f(x)=\frac12$ for $0\le x\le\pi$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   It is claimed that $\cos\theta+\sin\theta\equiv\sqrt2\cos\left(\theta-\frac{\pi}{4}\right)$ for all values of $\theta$.

   (a)

   Prove that the claim is true.

   [3 marks]

   (b)

   Hence find the greatest value of $\frac{1}{3+\cos\theta+\sin\theta}$, and the smallest positive value of $\theta$ at which it occurs.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A ball is projected from level ground with speed $u$ m s$^{-1}$ at an angle $\theta$ above the horizontal, where $0<\theta<90^{\circ}$. Air resistance is negligible and $g=9.8$ m s$^{-2}$.

   (a)

   Show that the horizontal range of the ball is $\frac{u^2\sin2\theta}{g}$.

   [6 marks]

   (b)

   The ball is projected at $u=20$ and lands 30 m from the point of projection. (i) Find the two possible values of $\theta$, to 3 significant figures. (ii) Explain why the two values of $\theta$ always sum to $90^{\circ}$. (iii) Find the greatest range possible with this speed.

   [6 marks]

   Total for question 4: 12 marks

End of questions