Question

.. \(\cos \left((2 n+1) \frac{\pi}{2}+\theta\right)\) is given by (a) \((-1)^{\mathrm{n}} \cos \theta\) (b) \((-1)^{\mathrm{n}-1} \cos \theta\) (c) \((-1)^{\mathrm{n}} \sin \theta\) (d) \((-1)^{\mathrm{n}+1} \sin \theta\)

Short answer

Answer: The expression for \(\cos \left((2 n+1) \frac{\pi}{2}+\theta\right)\) is \((-1)^{n}\cos \theta \cdot \mathbb{I}_{odd}(n) + (-1)^{n}\sin \theta \cdot \mathbb{I}_{even}(n)\), where \(\mathbb{I}_{odd}(n)\) and \(\mathbb{I}_{even}(n)\) are indicator functions returning 1 if \(n\) is odd or even, respectively, and 0 otherwise.

Step-by-step solution

Write the given expression in terms of angle sum formula

We are given the expression \(\cos \left((2 n+1) \frac{\pi}{2}+\theta\right)\). Let \(A = (2n+1)\frac{\pi}{2}\) and \(B = \theta\). Now we can write the expression as \(\cos (A+B)\).

Apply the angle sum formula

Now, we'll apply the angle sum formula \(\cos (A+B) = \cos A \cos B - \sin A \sin B\). We get: \(\cos ((2n+1) \frac{\pi}{2}+ \theta) = \cos((2n+1)\frac{\pi}{2})\cos \theta - \sin((2n+1)\frac{\pi}{2}) \sin \theta\).

Observe the patterns of sine and cosine functions

For \((2n+1)\frac{\pi}{2}\), note that when \(n\) is even \((2n+1)\frac{\pi}{2}\) is an odd multiple of \(\frac{\pi}{2}\) and when \(n\) is odd it is an even multiple of \(\frac{\pi}{2}\). Now, let's analyze the sine and cosine values for these cases: 1. When \(n\) is even, \((2n+1)\frac{\pi}{2}\) is an odd multiple of \(\frac{\pi}{2}\). In this case, \(\cos((2n+1)\frac{\pi}{2}) = 0\) and \(\sin((2n+1)\frac{\pi}{2}) = (-1)^{n}\). 2. When \(n\) is odd, \((2n+1)\frac{\pi}{2}\) is an even multiple of \(\frac{\pi}{2}\). In this case, \(\cos((2n+1)\frac{\pi}{2}) = (-1)^{n}\) and \(\sin((2n+1)\frac{\pi}{2}) = 0\).

Substitute the sine and cosine values and simplify

Using the information from step 3, we can substitute the sine and cosine values into the expression from step 2. Let's consider both cases: 1. When \(n\) is even: \(\cos((2n+1)\frac{\pi}{2})\cos \theta - \sin((2n+1)\frac{\pi}{2}) \sin \theta = 0\cos \theta - (-1)^{n} \sin \theta\). 2. When \(n\) is odd: \(\cos((2n+1)\frac{\pi}{2})\cos \theta - \sin((2n+1)\frac{\pi}{2}) \sin \theta = (-1)^{n}\cos \theta - 0\sin \theta\). In both cases, we can combine the expressions: \(\cos((2n+1)\frac{\pi}{2}+\theta) = (-1)^{n}\cos \theta \cdot \mathbb{I}_{odd}(n) + (-1)^{n}\sin \theta \cdot \mathbb{I}_{even}(n)\), where \(\mathbb{I}_{odd}(n)\) and \(\mathbb{I}_{even}(n)\) are indicator functions returning 1 if \(n\) is odd or even, respectively, and 0 otherwise.

Compare with the given options

Now, let's compare our result with the given options: (a) \((-1)^{\mathrm{n}} \cos \theta\) (b) \((-1)^{\mathrm{n}-1} \cos \theta\) (c) \((-1)^{\mathrm{n}} \sin \theta\) (d) \((-1)^{\mathrm{n}+1} \sin \theta\) Our simplified expression is a combination of option (a) and option (c), for odd and even values of \(n\), respectively: \(\cos ((2n+1)\frac{\pi}{2}+\theta) = (-1)^{n}\cos \theta \cdot \mathbb{I}_{odd}(n) + (-1)^{n}\sin \theta \cdot \mathbb{I}_{even}(n)\). Thus, there is no single expression that fits the given options, but our final expression remains correct.

Key concepts

Angle Sum Formula

Understanding the angle sum formula is a crucial part of trigonometry, often used in problems concerning the addition and subtraction of angles. Essentially, the angle sum formula provides us with a direct way to calculate the sine and cosine of the sum or difference of two angles.

Specifically, the angle sum formulas are as follows:

• For sine: \( \text{sin}(A+B) = \text{sin}A\text{cos}B + \text{cos}A\text{sin}B \)
• For cosine: \( \text{cos}(A+B) = \text{cos}A\text{cos}B - \text{sin}A\text{\text{sin}}B \)

These formulas enable us to convert a trigonometric function of a sum into a product of two functions, simplifying complex expressions. In the solution provided, you'll notice how the angle sum formula for cosine is applied to determine the value of \( \text{cos}((2n+1)\frac{\text{π}}{2}+θ) \). When working on related problems, remember that mastering the angle sum formula can immensely aid in solving various trigonometric equations.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent, are defined with respect to a right-angled triangle or the unit circle. They are fundamental in connecting angles to ratios of side lengths, and their significance extends to many areas including physics and engineering.

The primary trigonometric functions are:

• Sine (sin)
• Cosine (cos)
• Tangent (tan)

Each function has a different interpretation on the unit circle, with sine representing the y-coordinate, cosine the x-coordinate, and tangent the ratio of sine to cosine at a given angle. Furthermore, for angles that are multiples of \( \frac{\text{π}}{2} \), we can predict their sine and cosine values, as these points lie on the axes of the unit circle. These functions show specific patterns of behavior, which is highlighted in the problem's step-by-step solution by observing the sine and cosine values for odd and even multiples of \( \frac{\text{π}}{2} \).

Sine and Cosine Patterns

Recognizing the sine and cosine patterns, especially at multiples of \( \frac{π}{2} \), is highly beneficial for simplifying trigonometric expressions. These patterns follow from the unit circle, where the sine and cosine functions exhibit periodic behavior and specific values at certain intervals.

Here are some valuable points to note:

• \( \text{sin}(nπ) = 0 \) and \( \text{cos}(nπ) = (-1)^n \) for any integer n.
• \( \text{sin}\left(\frac{π}{2} + nπ\right) = (-1)^n \) and \( \text{cos}\left(\frac{π}{2} + nπ\right) = 0 \) for any integer n.

With these patterns, it becomes straightforward to evaluate trigonometric expressions at special angles. For example, in the given solution, when evaluating \( \text{cos}\left((2n+1)\frac{π}{2}+θ\right) \), we can deduce that for any integer n, the cosine part of the expression will be 0, as \( (2n+1)\frac{π}{2} \) is an odd multiple of \( \frac{π}{2} \). This knowledge is crucial for simplifying the given trigonometric expression.

Mathematical Induction

Mathematical induction is a powerful proof technique used in mathematics, especially to prove statements about natural numbers. It consists of two main steps:

• The base case: This is where you prove that the statement holds for the initial number, usually 0 or 1.
• The induction step: Here you assume the statement holds for an arbitrary natural number k and then show that its truth implies the statement must also hold for k+1.

Once these steps are established, it confirms that the statement is true for all natural numbers. While mathematical induction was not directly applied in this solution, it is a tool that can often be used to generalize patterns noticed in specific cases of trigonometric functions. For instance, once a pattern is observed for particular values of n, as in the sine and cosine values for multiples of \( \frac{π}{2} \), induction can sometimes be used to prove these observations for all integers n.