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Consider the complex number \(z=e^{i \theta}=\cos \theta+i \sin \theta .\) (a) By evaluating \(z^{2}\) two different ways, prove the trig identities \(\cos 2 \theta=\cos ^{2} \thetaQEDQED-\sin ^{2} \theta\) and \(\sin 2 \theta=2 \sin \theta \cos \theta .\) (b) Use the same technique to find corresponding identities for \(\cos 3 \theta\) and \(\sin 3 \theta\).

Short Answer

Expert verified
Use Euler's formula and equate real/imaginary components.

Step by step solution

01

Evaluate \( z^2 \) directly from exponential form

Since \( z = e^{i\theta} \), we raise it to the power of 2: \( z^2 = (e^{i\theta})^2 = e^{2i\theta} \). By Euler's formula, we know \( e^{2i\theta} = \cos(2\theta) + i\sin(2\theta) \).
02

Evaluate \( z^2 \) using trigonometric form

Given that \( z = \cos \theta + i \sin \theta \), we can calculate \( z^2 = (\cos \theta + i \sin \theta)^2 \). Using the identity \((a+b)^2 = a^2 + 2ab + b^2\), we expand it: \[ z^2 = \cos^2 \theta + 2i \cos \theta \sin \theta - \sin^2 \theta \].
03

Equate real parts to find \( \cos 2\theta \)

The real part of \( z^2 = \cos^2 \theta - \sin^2 \theta \) from the second method should equal the real part from Step 1, \( \cos 2\theta \). Thus, \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \).
04

Equate imaginary parts to find \( \sin 2\theta \)

The imaginary part of \( z^2 \) is \( 2i \cos \theta \sin \theta \) from the trigonometric form, which equals the imaginary part from Step 1, \( i \sin 2\theta \). Therefore, \( \sin 2\theta = 2 \sin \theta \cos \theta \).
05

Find \( z^3 \) using exponential form

For part (b), consider \( z^3 = (e^{i\theta})^3 = e^{3i\theta} \). By Euler’s formula, \( e^{3i\theta} = \cos(3\theta) + i\sin(3\theta) \).
06

Find \( z^3 \) using trigonometric expansion

We can expand \( z^3 = (\cos \theta + i \sin \theta)^3 \) using the binomial theorem. Calculate the terms: \[ z^3 = \cos^3 \theta + 3i \cos^2 \theta \sin \theta - 3 \cos \theta \sin^2 \theta - i \sin^3 \theta \].
07

Simplify and equate real parts for \( \cos 3\theta \)

Group the real parts: \( \cos^3 \theta - 3\cos \theta \sin^2 \theta \). Equating it to the real part from Step 5, \( \cos 3\theta \): \( \cos 3\theta = \cos^3 \theta - 3\cos \theta \sin^2 \theta \).
08

Simplify and equate imaginary parts for \( \sin 3\theta \)

Group the imaginary parts: \( 3\cos^2 \theta \sin \theta - \sin^3 \theta \) and equate it to the imaginary part from Step 5, \( \sin 3\theta \): \( \sin 3\theta = 3\cos^2 \theta \sin \theta - \sin^3 \theta \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Numbers
Complex numbers are numbers of the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit, defined by \( i^2 = -1 \). The real part is \( a \) and the imaginary part is \( b \). Complex numbers extend the idea of one-dimensional real numbers to two-dimensional numbers by representing them on a plane.

The complex plane, also known as the Argand plane, is a two-dimensional plane where the x-axis represents the real part and the y-axis represents the imaginary part of complex numbers. Each point on this plane corresponds to a complex number. The formula \( z = e^{i\theta} = \cos \theta + i \sin \theta \), used in the exercise above, is a well-known representation of complex numbers that connects them to trigonometric functions.

A crucial aspect of complex numbers is their modulus and argument. The modulus is the distance from the origin to the point in the complex plane and is calculated via \( |z| = \sqrt{a^2 + b^2} \). The argument is the angle formed with the positive real axis, often denoted as \( \theta \), calculated using trigonometric functions. Complex numbers provide a powerful toolset for solving equations that do not have real solutions and play a critical role in advanced mathematics and engineering.
Euler's Formula
Euler's formula is one of the most remarkable equations in mathematics. It states that for any real number \( \theta \), \( e^{i\theta} = \cos \theta + i \sin \theta \). This formula elegantly bridges the fields of trigonometry and complex analysis.

The formula provides a way to express complex numbers in polar form, making calculations involving complex numbers more manageable. Specifically, through Euler's formula, the exponential function \( e^{i\theta} \) links exponential growth to trigonometric oscillation, describing how complex numbers rotate around the origin in the complex plane.

Using Euler's formula, we can derive several trigonometric identities, as demonstrated in the exercise with \( z^{2} = e^{2i\theta} \), leading to the identities \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \) and \( \sin 2\theta = 2\sin \theta \cos \theta \). This not only shows how trigonometric identities can be derived from complex exponentials but highlights the utility of Euler’s formula in simplifying complex calculations.

This blending of trigonometry and complex numbers simplifies many mathematical problems, offering insights into the behavior of functions and phenomena that are inherently periodic or oscillatory.
Binomial Theorem
The binomial theorem provides a formula for expanding powers of sums. Specifically, it states that \((a + b)^n = \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k\), where \( \binom{n}{k} \) is a binomial coefficient. In our exercise, this theorem allows us to expand \((\cos \theta + i \sin \theta)^n\) to find expressions for higher powers of complex numbers.

When applying the binomial theorem to trigonometric forms of complex numbers, each term in the expansion represents a combination of real and imaginary parts raised to different powers. The theorem incorporates the idea of permutations of terms and coefficients, allowing a structured way to expand polynomials.

In the context of this exercise, using the binomial theorem helps derive formulas for \( \cos 3\theta \) and \( \sin 3\theta \). By expanding \( (\cos \theta + i \sin \theta)^3 \) and separating real and imaginary components, we obtain new trigonometric identities.

The binomial theorem is a powerful computational tool, not only facilitating the multiplication of expressions but also giving rise to identities across various branches of mathematics.

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Most popular questions from this chapter

2.16 \(\star\) A golfer hits his ball with speed \(v_{\mathrm{o}}\) at an angle \(\theta\) above the horizontal ground. Assuming that the angle \(\theta\) is fixed and that air resistance can be neglected, what is the minimum speed \(v_{\mathrm{o}}(\min )\) for which the ball will clear a wall of height \(h\), a distance \(d\) away? Your solution should get into trouble if the angle \(\tan \theta < h / d .\) Explain. What is \(v_{\mathrm{o}}(\min )\) if \(\theta=25^{\circ}, d=50 \mathrm{m},\) and \(h=2 \mathrm{m} ?\)

(a) Using Euler's relation (2.76), prove that any complex number \(z=x+\) iy can be written in the form \(z=r e^{i \theta},\) where \(r\) and \(\theta\) are real. Describe the significance of \(r\) and \(\theta\) with reference to the complex plane. (b) Write \(z=3+4 i\) in the form \(z=r e^{i \theta} .\) (c) Write \(z=2 e^{-i \pi / 3}\) in the form \(x+i y.\)

Use the series definition (2.72) of \(e^{z}\) to prove that \(^{12} d e^{z} / d z=e^{z}\).

For any complex number \(z=x+i y,\) the real and imaginary parts are defined as the real numbers \(\operatorname{Re}(z)=x\) and \(\operatorname{Im}(z)=y .\) The modulus or absolute value is \(|z|=\sqrt{x^{2}+y^{2}}\) and the phase or angle is the value of \(\theta\) when \(z\) is expressed as \(z=r e^{i \theta} .\) The complex conjugate is \(z^{*}=x-i y\) (This last is the notation used by most physicists; most mathematicians use \(\bar{z}\).) For each of the following complex numbers, find the real and imaginary parts, the modulus and phase, and the complex conjugate, and sketch \(z\) and \(z^{*}\) in the complex plane: (a) \(z=1+i \quad\) (b) \(z=1-i \sqrt{3}\) (d) \(z=5 e^{i \omega t}\) (c) \(z=\sqrt{2} e^{-i \pi / 4}\) In part (d), \(\omega\) is a constant and \(t\) is the time.

Consider an object that is coasting horizontally (positive \(x\) direction) subject to a drag force \(f=-b v-c v^{2} .\) Write down Newton's second law for this object and solve for \(v\) by separating variables. Sketch the behavior of \(v\) as a function of \(t .\) Explain the time dependence for \(t\) large. (Which force term is dominant when \(t\) is large?)

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