Integration using De Moivre identities

Pre-U Pre-U 9795/1 Specimen Q11

14 marks Challenging +1.2

11 The complex number $z$ is defined as $z = \cos \theta + \mathrm { i } \sin \theta$.

Show that $z ^ { n } + z ^ { - n } = 2 \cos n \theta$.
By expanding $\left( z + z ^ { - 1 } \right) ^ { 5 }$, show that $16 \cos ^ { 5 } \theta = \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta$.
Hence find $\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { 5 } \theta \mathrm {~d} \theta$.
Sketch the graphs of $\mathrm { f } ( \theta ) = \sin ^ { 5 } \theta$ and $\mathrm { f } ( \theta ) = \cos ^ { 5 } \theta$, for $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$, and hence give the value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { 5 } \theta \mathrm {~d} \theta$$

CAIE FP1 2003 November Q8

11 marks Challenging +1.2

Given that $z = e^{i\theta}$ and $n$ is a positive integer, show that $$z^n + \frac{1}{z^n} = 2 \cos n\theta \quad \text{and} \quad z^n - \frac{1}{z^n} = 2i \sin n\theta.$$ [2] Hence express $\sin^6 \theta$ in the form $$p \cos 6\theta + q \cos 4\theta + r \cos 2\theta + s,$$ where the constants $p$, $q$, $r$, $s$ are to be determined. [4] Hence find the mean value of $\sin^6 \theta$ with respect to $\theta$ over the interval $0 \leq \theta \leq \frac{1}{4}\pi$. [5]

CAIE FP1 2018 November Q8

10 marks Challenging +1.3

By considering the binomial expansion of $\left(z + \frac{1}{z}\right)^6$, where $z = \cos \theta + \mathrm{i} \sin \theta$, express $\cos^6 \theta$ in the form $$\frac{1}{32}(p + q \cos 2\theta + r \cos 4\theta + s \cos 6\theta),$$ where $p, q, r$ and $s$ are integers to be determined. [6]
Hence find the exact value of $$\int_{-\frac{1}{4}\pi}^{\frac{1}{4}\pi} \cos^6\left(\frac{1}{2}x\right) \mathrm{d}x.$$ [4]

CAIE Further Paper 2 2020 June Q8

15 marks Challenging +1.8

Use de Moivre's theorem to show that $\sin^6 \theta = -\frac{1}{32}(\cos 6\theta - 6\cos 4\theta + 15\cos 2\theta - 10)$. [6]

It is given that $\cos^6 \theta = \frac{1}{32}(\cos 6\theta + 6\cos 4\theta + 15\cos 2\theta + 10)$.

Find the exact value of $\int_0^{\frac{1}{4}\pi}\left(\cos^6\left(\frac{1}{4}x\right) + \sin^6\left(\frac{1}{4}x\right)\right)dx$. [4]
Express each root of the equation $16c^6 + 16\left(1-c^2\right)^3 - 13 = 0$ in the form $\cos k\pi$, where $k$ is a rational number. [5]

OCR MEI FP2 2011 January Q2

19 marks Standard +0.3

1. Given that $z = \cos \theta + j \sin \theta$, express $z^n + z^{-n}$ and $z^n - z^{-n}$ in simplified trigonometrical form. [2]
2. By considering $(z + z^{-1})^6$, show that $$\cos^6 \theta = \frac{1}{32}(\cos 6\theta + 6 \cos 4\theta + 15 \cos 2\theta + 10).$$ [3]
3. Obtain an expression for $\cos^6 \theta - \sin^6 \theta$ in terms of $\cos 2\theta$ and $\cos 6\theta$. [5]
The complex number $w$ is $8e^{i\pi/3}$. You are given that $z_1$ is a square root of $w$ and that $z_2$ is a cube root of $w$. The points representing $z_1$ and $z_2$ in the Argand diagram both lie in the third quadrant.
1. Find $z_1$ and $z_2$ in the form $re^{i\theta}$. Draw an Argand diagram showing $w$, $z_1$ and $z_2$. [6]
2. Find the product $z_1z_2$, and determine the quadrant of the Argand diagram in which it lies. [3]

OCR FP3 Q4

8 marks Challenging +1.2

The integrals $C$ and $S$ are defined by $$C = \int_0^{2\pi} e^{3x} \cos 3x \, dx \quad \text{and} \quad S = \int_0^{2\pi} e^{3x} \sin 3x \, dx.$$ By considering $C + iS$ as a single integral, show that $$C = \frac{1}{13}(2 + 3e^\pi),$$ and obtain a similar expression for $S$. [8] (You may assume that the standard result for $\int e^{kx} dx$ remains true when $k$ is a complex constant, so that $\int e^{(a+ib)x} dx = \frac{1}{a+ib} e^{(a+ib)x}$.)

OCR FP3 Q8

12 marks Standard +0.8

By expressing $\sin \theta$ in terms of $e^{i\theta}$ and $e^{-i\theta}$, show that $$\sin^6 \theta \equiv \frac{1}{32}(\cos 6\theta - 6\cos 4\theta + 15\cos 2\theta - 10).$$ [5]
Replace $\theta$ by $\left(\frac{1}{2}\pi - \theta\right)$ in the identity in part (i) to obtain a similar identity for $\cos^6 \theta$. [3]
Hence find the exact value of $\int_0^{2\pi} \left(\sin^6 \theta - \cos^6 \theta\right) d\theta$. [4]

OCR FP3 2008 January Q4

8 marks Standard +0.8

The integrals $C$ and $S$ are defined by $$C = \int_0^{3\pi} e^{3x} \cos 3x \, dx \quad \text{and} \quad S = \int_0^{3\pi} e^{3x} \sin 3x \, dx.$$ By considering $C + iS$ as a single integral, show that $$C = -\frac{1}{3}(2 + 3e^{\pi}),$$ and obtain a similar expression for $S$. [8] (You may assume that the standard result for $\int e^{kx} dx$ remains true when $k$ is a complex constant, so that $\int e^{(a+ib)x} dx = \frac{1}{a+ib}e^{(a+ib)x}$.)

OCR FP3 2011 January Q3

8 marks Standard +0.3

Express $\sin \theta$ in terms of $e^{i\theta}$ and $e^{-i\theta}$ and show that $$\sin^4 \theta \equiv \frac{1}{8}(\cos 4\theta - 4\cos 2\theta + 3).$$ [4]
Hence find the exact value of $\int_0^{\frac{\pi}{4}} \sin^4 \theta \, d\theta$. [4]

AQA Further Paper 1 2019 June Q8

10 marks Standard +0.8

If $z = \cos \theta + i \sin \theta$, use de Moivre's theorem to prove that $$z^n - \frac{1}{z^n} = 2i \sin n\theta$$ [3 marks]
Express $\sin^5 \theta$ in terms of $\sin 5\theta$, $\sin 3\theta$ and $\sin \theta$ [4 marks]
Hence show that $$\int_0^{\frac{\pi}{3}} \sin^5 \theta \, d\theta = \frac{53}{480}$$ [3 marks]

AQA Further Paper 1 Specimen Q12

3 marks Challenging +1.8

The function $f(x) = \cosh(ix)$ is defined over the domain $\{x \in \mathbb{R} : -a\pi \leq x \leq a\pi\}$, where $a$ is a positive integer. By considering the graph of $y = [f(x)]^n$, find the mean value of $[f(x)]^n$, when $n$ is an odd positive integer. Fully justify your answer. [3 marks]