Question

Let \(z=\cos \theta+i \sin \theta\). Then the value of $$ \sum_{n=1}^{15} \operatorname{Im}\left(z^{2 m-1}\right) $$ at \(\theta=2^{\circ}\) is (A) \(\frac{1}{\sin 2^{\circ}}\) (B) \(\frac{1}{3 \sin 2^{\circ}}\) (C) \(\frac{1}{2 \sin 2^{\circ}}\) (D) \(\frac{1}{4 \sin 2^{\circ}}\)

Short answer

None of the provided answer choices match the solution derived from the provided sum. There might be an error in the problem statement or in the interpretation of the summation.

Step-by-step solution

Understand the nature of z

Recognize that the complex number z given by the form \(z = \cos\theta + i\sin\theta\) represents a point on the unit circle in the complex plane, also known as Euler's formula. This formula can be used to find powers of z.

Calculate z to the power of an odd integer

Since the exponent in the summand is in the form of \(2m - 1\), it's always going to be an odd integer. Recognize that \(z^{2m-1}\) can be similarly represented using Euler's formula: \(z^{2m-1} = \cos((2m-1)\theta) + i\sin((2m-1)\theta)\). The imaginary part \(\text{Im}(z^{2m-1})\) is therefore \(\sin((2m-1)\theta)\).

Set up the summation of the imaginary parts

As asked, the sum is \(\sum_{m=1}^{15} \text{Im}(z^{2m-1}) = \sum_{m=1}^{15} \sin((2m-1)\cdot 2^\circ)\). This sum includes the sin of angles: 2°, 6°, 10°, ..., 58°.

Recognize the pattern in the angles

Note that the angles in the sine function form an arithmetic sequence with a common difference of 4°. This is important because we can use properties of sine and addition angles to simplify the summation.

Express the summation terms using the sine of a double angle

Observe that each term in the series can be expressed as \(\sin(2n\cdot 1^\circ)\) where \(n\) varies from 1 to 29 skipping the even numbers. This allows us to pair up terms that add up to 30°: \(\sin(2^\circ) + \sin(58^\circ)\), ..., \(\sin(26^\circ) + \sin(34^\circ)\). Since \(\sin(\alpha) = \sin(180^\circ - \alpha)\), the sum of each pair is equal to \(\sin(60^\circ)\) or \(\frac{\sqrt{3}}{2}\).

Count the number of pairs and calculate the sum

There are 15 terms in the summation, which gives us 7 pairs and one single term of \(\sin(30^\circ)\). Thus the sum is \(7\cdot\frac{\sqrt{3}}{2} + \frac{1}{2}\).

Simplify the summation

The sum simplifies to \(\frac{7\sqrt{3}}{2} + \frac{1}{2} = \frac{7\sqrt{3} + 1}{2}\). Assuming there's an error in the summation range, and it should actually be a finite geometric series (since the series as given doesn't simplify to any of the answer choices), we would need to find a formula that fits the simplified sum with the given answer choices.

Correlate the result with the given options

None of the steps provided lead to a summation that coincides with the given answer choices. This suggests that there might be a mistake in the summation range or in the interpretation of the problem, as the standard approach to solving this does not yield an answer that matches the options (A) through (D). Further clarification or a review of the original problem statement may be required.

Key concepts

Euler's Formula

Euler's formula is a cornerstone of complex number theory and provides a link between trigonometry and complex exponentials. It's expressed as
\[ e^{i\theta} = \cos \theta + i \sin \theta \]
which shows how a complex number can be represented in trigonometric form. Here, \( e \) is the base of the natural logarithm, \( i \) is the imaginary unit, and \( \theta \) is the angle in radians. For students dealing with complex numbers, Euler's formula is invaluable as it simplifies multiplication and division, powers, and roots of complex numbers. It's also essential when converting between the standard rectangular coordinates (a + bi) and polar coordinates (r(cos θ + i sin θ)). When understanding summation of series with complex numbers in trigonometric form, Euler's formula can help break down each term into more manageable parts.

Summation of Series

The summation of series is a topic central to both calculus and algebra. It involves adding up a sequence of terms to find their total. In some cases, particularly when dealing with arithmetic or geometric progressions, there are formulas available that simplify the process of finding the sum.
For example, the sum of the first \( n \) terms of an arithmetic progression can be found using the formula:
\[ S_n = \frac{n}{2}(a_1 + a_n) \]
where \( a_1 \) is the first term and \( a_n \) is the nth term. The summation of complex numbers often involves taking the sum of their real parts and imaginary parts separately, which is crucial when analyzing series that involve powers of complex numbers.

Imaginary Part of Complex Numbers

The imaginary part of a complex number is the coefficient of the imaginary unit \( i \), where \( i^2 = -1 \). For a complex number written in the form \( a + bi \), the imaginary part is \( b \). In the context of Euler's formula, the imaginary part corresponds to the sine component.
Thus, when dealing with the complex number \( z = \cos \theta + i \sin \theta \), the imaginary part is \( \sin \theta \). Understanding this is critical when you are working with the summation of series of complex numbers raised to a power, as you would often be asked to find the sum of just the imaginary parts of these numbers, as seen in the exercise provided.

Arithmetic Progression

An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. This difference is often referred to as the 'common difference'. If \( a_1 \) represents the first term, \( d \) the common difference, the nth term \( a_n \) is given by:
\[ a_n = a_1 + (n - 1)d \]
In the context of the given exercise, the angles within the sine function demonstrate an arithmetic progression with a common difference of \( 4^\circ \). This insight is significant for simplifying the computation of the series sum, as it aids in identifying patterns and potential symmetries within the sequence of angles, especially when they are used within trigonometric functions such as sine. By mastering arithmetic progressions, students can better tackle problems involving the summation of series and apply shortcuts to find sums more efficiently.