Harmonic form and trigonometric proofEdexcel A-Level Maths: Subtopic test

1. 1
   Let $h(\theta)=7\cos\theta-24\sin\theta$, where $\theta$ is measured in degrees.

   (a)

   Write $h(\theta)$ in the form $R\cos(\theta+\alpha)$, where $R>0$ and $\alpha$ is acute. Find $R$.

   [1 mark]

   • A$31$
   • B$17$
   • C$25$
   • D$625$

   (b)

   For $h(\theta)=R\cos(\theta+\alpha)$ with $R>0$ and $\alpha$ acute, which is the value of $\tan\alpha$?

   [1 mark]

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

   (c)

   Find the maximum value of $h(\theta)$ and the smallest positive value of $\theta$ at which it occurs, giving $\theta$ to 1 decimal place.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider the expression $E=\cos x\cos2x+\sin x\sin2x$.

   (a)

   Which expression is equal to $E$?

   [1 mark]

   • A$\cos(2x-x)$
   • B$\cos3x$
   • C$\sin(2x-x)$
   • D$\sin3x$

   (b)

   Which expression is equal to $\sin x\cos2x-\cos x\sin2x$?

   [1 mark]

   • A$\sin x$
   • B$\sin3x$
   • C$-\sin3x$
   • D$-\sin x$

   (c)

   Prove that $E\equiv\cos x$, and hence solve $E=\frac12$ for $0\le x<360^\circ$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the identity $\dfrac{\sin2\theta}{1+\cos2\theta}\equiv\tan\theta$.

   (a)

   Prove that the identity is true.

   [3 marks]

   (b)

   Hence solve $\dfrac{\sin2\theta}{1+\cos2\theta}=3\tan^2\theta$ for $0\le\theta<360^\circ$, giving answers to 1 decimal place where necessary.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $f(x)=\sin x-\sqrt3\cos x$, where $x$ is measured in radians.

   (a)

   (i) Express $f(x)$ in the form $R\sin(x-\alpha)$, where $R>0$ and $0<\alpha<\frac\pi2$, giving $\alpha$ exactly.
   (ii) Hence solve $f(x)=1$ for $0\le x<2\pi$, giving exact answers.

   [6 marks]

   (b)

   (i) Write down the minimum value of $f(x)$ and the smallest positive value of $x$ at which it occurs.
   (ii) Hence find the greatest and least values of $g(x)=\dfrac{12}{4+f(x)}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions