Problem 58 - Algebra and Trigonometry

Question

If \(f(x)=\sin x\) and \(f(a)=\frac{1}{4},\) find the value of $$ f(a)+f(a+2 \pi)+f(a+4 \pi)+f(a+6 \pi) $$

Step-by-Step Solution

Answer

The value of \(f(a) + f(a+2\pi) + f(a+4\pi) + f(a+6\pi)\) is 1.

Step 1: Recognize the periodic property of Sine function

Observe that \(f(x)=\sin x\) is a periodic function with a period of \(2\pi\). This means that for every \(x\), \(f(x + 2\pi) = f(x)\).

Step 2: Use the periodicity

Using the periodic property of the sine function, we can find that \(f(a+2\pi)=f(a)\), \(f(a+4\pi) = f(a)\), and \(f(a+6\pi) = f(a)\).

Step 3: Utilize given value and sum up

Substitute \(f(a) = \frac{1}{4}\) into above results, and sum all of them to get the total: \(f(a) + f(a+2\pi) + f(a+4\pi) + f(a+6\pi) = \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = 1.\)

Key Concepts

Sine Function

The sine function is one of the core trigonometric functions and is crucial in understanding periodic phenomena. It is commonly represented as \( f(x) = \sin x \). This function describes the y-coordinate of a point on the unit circle as that point travels around the circle.

• At \( x = 0 \), \( \sin x = 0 \).
• At \( x = \frac{\pi}{2} \), \( \sin x = 1 \).
• At \( x = \pi \), \( \sin x = 0 \).
• At \( x = \frac{3\pi}{2} \), \( \sin x = -1 \).
• It returns to zero at \( x = 2\pi \), completing one cycle.

This cyclical behavior makes the sine function ideal for modeling waves and other oscillating systems. The amplitude of the sine function is 1, meaning it oscillates between -1 and 1. Understanding the sine function's properties helps in solving advanced problems involving waves, sound, and light.

Periodicity

Periodicity is a fundamental property of some functions where they repeat their values at regular intervals. For the sine function, this repetitiveness occurs every \(2\pi\) units. Hence, the period of \( \sin x \) is \( 2\pi \). This means the sine function's graph will look the same between \( x = 0 \) and \( x = 2\pi \), \( x = 2\pi \) and \( x = 4\pi \), and so forth. The periodic nature of sine allows you to make useful simplifications:

• \( \sin(x + 2\pi) = \sin x \)
• This property is due to the circular nature of trigonometric functions, reflecting the consistent cycle of angles on a circle.

In the given exercise, we used periodicity to show that \( f(a) = f(a + 2\pi) = f(a + 4\pi) = f(a + 6\pi) \). Understanding periodicity is crucial when working with functions that model cyclic processes, such as tides, sound waves, or seasonal patterns.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are valid for any value of the involved variable. These identities are essential tools in trigonometry to simplify complex expressions and solve trigonometric equations. There are several fundamental identities to be aware of:

• Pythagorean Identity: \( \sin^2 x + \cos^2 x = 1 \).
• Angle sum and difference identities, such as \( \sin(x \pm y) = \sin x \cos y \pm \cos x \sin y \).
• Double angle identities, for instance, \( \sin 2x = 2 \sin x \cos x \).

These identities can help transform and solve equations involving sine (\( \sin x \)) and other trigonometric functions. Although the original problem mainly focuses on periodicity, knowing these identities can further facilitate complex problem-solving in trigonometry, helping you reduce and analyze expressions efficiently.