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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

[Computer] Use suitable graph-plotting software to plot graphs of the trajectory (2.36) of a projectile thrown at 45 ^above the horizontal and subject to linear air resistance for four different values of the drag coefficient, ranging from a significant amount of drag down to no drag at all. Put all four trajectories on the same plot. [Hint: In the absence of any given numbers, you may as well choose convenient values. For example, why not take \(v_{x \mathrm{o}}=v_{y \mathrm{o}}=1\) and \(g=1 .\) (This amounts to choosing your units of length and time so that these parameters have the value \(1 .\).) With these choices, the strength of the drag is given by the one parameter \(v_{\text {ter }}=\tau,\) and you might choose to plot the trajectories for \(v_{\text {ter }}=0.3,1,3,\) and \(\infty\) (that is, no drag at all), and for times from \(t=0\) to \(3 .\) For the case that \(v_{\text {ter }}=\infty\) you'll probably want to write out the trajectory separately.]

The hyperbolic functions cosh \(z\) and \(\sinh z\) are defined as follows: $$\cosh z=\frac{e^{z}+e^{-z}}{2} \quad \text { and } \quad \sinh z=\frac{e^{z}-e^{-z}}{2}$$ for any \(z,\) real or complex. (a) Sketch the behavior of both functions over a suitable range of real values of \(z\). (b) Show that \(\cosh z=\cos (i z)\). What is the corresponding relation for \(\sinh z ?\) (c) What are the derivatives of cosh \(z\) and \(\sinh z ?\) What about their integrals? ( \(\mathbf{d}\) ) Show that \(\cosh ^{2} z-\sinh ^{2} z=1\) (e) Show that \(\int d x / \sqrt{1+x^{2}}=\operatorname{arcsinh} x\). [Hint: One way to do this is to make the substitution \(x=\sinh z .]\)

I kick a puck of mass \(m\) up an incline (angle of slope \(=\theta\) ) with initial speed \(v_{\mathrm{o}}\). There is no friction between the puck and the incline, but there is air resistance with magnitude \(f(v)=c v^{2} .\) Write down and solve Newton's second law for the puck's velocity as a function of \(t\) on the upward journey. How long does the upward journey last?

Consider a projectile launched with velocity \(\left(v_{x 0}, v_{y_{0}}\right)\) from horizontal ground (with \(x\) measured horizontally and \(y\) vertically up). Assuming no air resistance, find how long the projectile is in the air and show that the distance it travels before landing (the horizontal range) is \(2 v_{x_{0}} v_{y_{0}} / g.\)

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

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