Problem 79 - Algebra and Trigonometry

Question

Use an identity to solve each equation on the interval \([0,2 \pi)\) $$ \sin \left(x+\frac{\pi}{4}\right)+\sin \left(x-\frac{\pi}{4}\right)=1 $$

Step-by-Step Solution

Answer

The solutions to the equation are \( x=\frac{\pi}{4} \) and \( x=\frac{3\pi}{4} \)

Step 1: Applying Sum-to-Product Identities

Using the trigonometric identities \( \sin(a) + \sin(b) = 2 \sin(\frac{a+b}{2}) \cos(\frac{a-b}{2}) \), we can rewrite the given equation \(\sin \left(x+\frac{\pi}{4}\right) + \sin \left(x-\frac{\pi}{4}\right) = 1\) as \(2 \sin(x) \cos(\frac{\pi}{4}) = 1\)

Step 2: Simplifying the equation

As \( \cos(\frac{\pi}{4}) \) is \( \frac{\sqrt{2}}{2} \), substituting this gives the equation as \(2 \sin(x) \frac{\sqrt{2}}{2} = 1\), which simplifies to \( \sin(x) = \frac{\sqrt{2}}{2} \)

Step 3: Solving for x

Within the interval \([0,2 \pi)\) the sine function equals \( \frac{\sqrt{2}}{2} \) at \( x=\frac{\pi}{4} \) and \( x=\frac{3\pi}{4} \). Therefore, the solutions to this equation are \( x=\frac{\pi}{4} \) and \( x=\frac{3\pi}{4} \)

Key Concepts

Sum-to-Product Identities

The sum-to-product identities are incredibly useful when simplifying trigonometric expressions. They allow us to transform the sum of two sine or cosine functions into a product. This can make solving equations much more straightforward.
For example, the identity \( \sin(a) + \sin(b) = 2 \sin\left(\frac{a+b}{2}\right) \cos\left(\frac{a-b}{2}\right) \) helps us here. It turns a sum of sines into a product of sine and cosine.
In the exercise, applying this identity transforms \( \sin \left( x+\frac{\pi}{4} \right) + \sin \left( x-\frac{\pi}{4} \right) \) into \( 2 \sin(x) \cos\left(\frac{\pi}{4}\right) \).
This conversion uses the angles \( a = x+\frac{\pi}{4} \) and \( b = x-\frac{\pi}{4} \), simplifying the solving process.

Sine Function

The sine function is a fundamental part of trigonometry, often represented as \( \sin(x) \). It describes a wave-like pattern, oscillating between -1 and 1. This periodic behavior repeats every \( 2\pi \) radians.
The sine function is useful because it applies to real-world scenarios like sound waves and circular motion. In our equation, we focus on finding when \( \sin(x) = \frac{\sqrt{2}}{2} \).
Understanding this value is key because it relates to well-known angles that frequently appear in trigonometry.

• \( \sin(x) = \frac{\sqrt{2}}{2} \) at \( x = \frac{\pi}{4} \) and \( x = \frac{3\pi}{4} \).

These are the points within \([0, 2\pi)\) where the sine function reaches this value.

Trigonometric Equations

Trigonometric equations involve unknowns represented by angles. Solving these equations often requires identities and transformations to simplify them.
In our problem, we started with \( \sin \left( x+\frac{\pi}{4} \right) + \sin \left( x-\frac{\pi}{4} \right) = 1 \). Applying the sum-to-product identities turned it into a simpler form.
By substituting known values, like \( \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \), the equation became \( \sin(x) = \frac{\sqrt{2}}{2} \).

• This final form is much easier to solve, resulting in the well-known angles \( x = \frac{\pi}{4} \) and \( x = \frac{3\pi}{4} \) within the given interval.

The process illustrates how breaking down complex equations into known identities simplifies the solution.

Angle Addition Formulas

Angle addition formulas are valuable tools in trigonometry. They allow us to express trigonometric functions of angle sums or differences in terms of functions of single angles.
For example, \( \sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b) \). However, in this exercise, we didn't need to directly apply these formulas.
Instead, understanding these principles helps reinforce why identities, like sum-to-product, work effectively. They provide a foundation for transforming and solving equations with combined angles, aiding our comprehension of how angles combine and influence trigonometric expressions efficiently.