Question

Prove that $$ \begin{aligned} &\cos \theta+\cos 3 \theta+\cos 5 \theta+\cdots+\cos (2 n-1) \theta=\frac{\sin 2 n \theta}{2 \sin \theta} \\ &\sin \theta+\sin 3 \theta+\sin 5 \theta+\cdots+\sin (2 n-1) \theta=\frac{\sin ^{2} n \theta}{\sin \theta} \end{aligned} $$ Hint : Use Euler's formula and the geometric progression formula.

Short answer

The cosine and sine series simplify to \(\frac{\sin 2n\theta}{2\sin\theta}\) and \(\frac{\sin^2 n\theta}{\sin\theta}\), respectively.

Step-by-step solution

Understand Euler's Formula

Euler's formula states that for any real number \theta, \[e^{i\theta} = \text{cos} \theta + i \text{sin} \theta\]This will help us convert trigonometric expressions to exponential form.

Convert Given Series Using Euler's Formula

Express the cosine and sine sums using Euler's formula. For the cosine series: \[\text{cos} \theta + \text{cos} 3\theta + \text{cos} 5\theta + \text{cos} (2n-1)\theta\]Rewrite each term using \[\text{cos} k\theta = \frac{e^{ik\theta} + e^{-ik\theta}}{2}\]. For the sine series: \[\text{sin} \theta + \text{sin} 3\theta + \text{sin} 5\theta + \text{sin} (2n-1)\theta\]Rewrite each term using \[\text{sin} k\theta = \frac{e^{ik\theta} - e^{-ik\theta}}{2i}\].

Derive the General Term

Sum the exponentials. For the cosine series, sum \[\frac{e^{i\theta} + e^{-i\theta}}{2} + \frac{e^{i3\theta} + e^{-i3\theta}}{2} + \text{...} + \frac{e^{i(2n-1)\theta} + e^{-i(2n-1)\theta}}{2}\]. For the sine series, sum \[\frac{e^{i\theta} - e^{-i\theta}}{2i} + \frac{e^{i3\theta} - e^{-i3\theta}}{2i} + \text{...} + \frac{e^{i(2n-1)\theta} - e^{-i(2n-1)\theta}}{2i}\].

Sum the Geometric Progression

Recognize the geometric series in the exponential sums. For cosine: \[\frac{1}{2}\re\bigg\frac{e^{i\theta}(1 - e^{i(2n)\theta})}{1 - e^{i2\theta}} + \frac{e^{-i\theta}(1 - e^{-i(2n)\theta})}{1 - e^{-i2\theta}}\bigg\]. For sine: \[\frac{1}{2i}\re\bigg\frac{e^{i\theta}(1 - e^{i(2n)\theta})}{1 - e^{i2\theta}} - \frac{e^{-i\theta}(1 - e^{-i(2n)\theta})}{1 - e^{-i2\theta}}\bigg\].

Simplify Expressions

Simplify the cosine series:\[\frac{1}{2} \frac{\text{sin}(2n\theta)}{\text{sin}(\theta)}\]. Simplify the sine series:\[\frac{1}{2 \text{sin} (\theta)} (2i \text{sin} (n\theta) \text{cos} (n\theta))\].

Final Expressions

Combine simplifications for cosine: \[\frac{\text{sin}(2n\theta)}{2 \text{sin} (\theta)}\]. Combine simplifications for sine: \[\frac{\text{sin}^2 (n \theta)}{\text{sin} (\theta)}\].

Key concepts

Euler's Formula

Euler's Formula is an essential mathematical tool that bridges trigonometry and complex numbers. It states that \(e^{i\theta} = \cos \theta + i \sin \theta\). This formula helps convert trigonometric functions into their exponential forms, making them easier to manipulate. When proving series summations in trigonometry, converting trigonometric functions into exponential forms helps streamline calculations. For instance, using Euler's formula, \( \cos \theta \) becomes \( \cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2} \), and \( \sin \theta \) becomes \( \sin \theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}. \) This transformation simplifies complex summations and prepares them for further algebraic operations such as geometric progressions.

Geometric Progression

A geometric progression (or geometric series) is a sequence of terms where each term is obtained by multiplying the previous term by a constant ratio. It has the general form \(a, ar, ar^2, ar^3, \ldots\). In trigonometric series summation, recognizing the structure of terms in geometric progression allows us to use established formulas to simplify summation. For example, the cosine series \( \cos \theta + \cos 3\theta + \cos 5\theta + \ldots\ + \cos (2n-1)\theta \) can be transformed into exponential form and summed using geometric progression formulas. The sum of a geometric series \( a + ar + ar^2 + \ldots + ar^{n-1} \) is given by \[ S = \frac{a(1 - r^n)}{1 - r} \]. Applying this within the context of Euler's formula and trigonometric identities enables the simplification of complex trigonometric series into more manageable forms.

Trigonometric Identities

Trigonometric identities are equations that relate different trigonometric functions to one another. They are crucial in transforming and simplifying expressions. Some key identities include:

• Sum and Difference Formulas: \[ \cos(a + b) = \cos a \cos b - \sin a \sin b, \quad \sin(a + b) = \sin a \cos b + \cos a \sin b \].
• Double Angle Formulas: \[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta, \quad \sin 2\theta = 2\sin \theta \cos \theta \].
• Half Angle Formulas: \[ \cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}, \quad \sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}} \].

In the provided exercise, these identities simplify intermediary steps and final results. For instance, using double angle formulas helps simplify expressions during the transition from exponential forms back to trigonometric sums by collaborating with Euler's formula.

Complex Numbers in Trigonometry

Complex numbers significantly enhance trigonometric and exponential calculations, as observed in Euler's formula. They generally take the form \( z = a + bi \), where \( i \) is the imaginary unit (\( i^2 = -1 \)). When dealing with series summation, the exponential form of complex numbers simplifies the process. For example, summing a series of sines and cosines can be done more efficiently by converting each cosine and sine term into its exponential form using Euler's formula. This approach leverages the properties of complex exponentials, simplifying what would be cumbersome trigonometric manipulations. By transforming the sums into geometric progressions involving complex exponentials, we can apply geometric series sum formulas and then revert back to trigonometric form, yielding simpler and clearer results.