Problem 41 - Algebra 2

Question

Verify each identity. $$ \sin \left(x+\frac{\pi}{3}\right)+\sin \left(x-\frac{\pi}{3}\right)=\sin x $$

Step-by-Step Solution

Answer

The given identity \(\sin \left(x+\frac{\pi}{3}\right)+\sin \left(x-\frac{\pi}{3}\right)=\sin x\) was verified by applying the addition and subtraction formulas for sine, and simplifying the resulting expression.

Step 1: Apply addition and subtraction formulas

The formula for \(\sin(a+b)\) is \(\sin a \cos b + \cos a \sin b\) and the formula for \(\sin(a-b)\) is \(\sin a \cos b - \cos a \sin b\). Apply these formulas to the given identity: \[\sin \left( x+\frac{\pi}{3} \right) + \sin \left( x-\frac{\pi}{3} \right) = \sin x \cos \frac{\pi}{3} + \cos x \sin \frac{\pi}{3} + \sin x \cos \frac{\pi}{3} - \cos x \sin \frac{\pi}{3}\]

Step 2: Simplify the result

Notice that two terms are identical and their sum is zero. This fact can be utilized to simplify the equation. The identity becomes: \[\sin x \cos \frac{\pi}{3} + \sin x \cos \frac{\pi}{3} = 2 \sin x \cos \frac{\pi}{3} = \sin x \]because \(\cos \frac{\pi}{3} = \frac{1}{2}\). This simplifies the identity to the right-hand side: \(\sin x\).

Key Concepts

Addition Formula

In trigonometry, the addition formula for the sine function helps to find the sine of a sum of two angles. The general form is given by:

• \( \sin(a+b) = \sin a \cos b + \cos a \sin b \)

This formula allows us to break down a complex trigonometric term into simpler parts, using the known values of the sine and cosine for individual angles. By applying this formula, you can transform expressions like \( \sin(x+\frac{\pi}{3}) \) into expressions involving \( \sin x \), \( \cos x \), and constants \( \cos \frac{\pi}{3} \) and \( \sin \frac{\pi}{3} \).
When tackling problems with addition formulas, it’s important to keep track of each part of the equation through

• the individual sine and cosine terms
• their respective angles

As you progress in solving these problems, carefully apply algebraic simplifications where possible to reach a simplified and more understandable form.

Subtraction Formula

Just like the addition formula helps with sums of angles, the subtraction formula assists in calculating the sine of a difference between two angles. The formula is given by:

• \( \sin(a-b) = \sin a \cos b - \cos a \sin b \)

This subtraction formula is handy for expressing any initial complex trigonometric problem in more manageable steps, just like how we applied it to simplify \( \sin(x-\frac{\pi}{3}) \) in the given exercise.
Usage of these formulas requires careful attention to algebraic details, as it involves properly combining and canceling terms. In the exercise, notice how terms \( \cos x \sin \frac{\pi}{3} \) in addition and subtraction cancelled each other. This insight is useful for eliminating redundant parts of the expressions, helping to reach the final solution cleanly.

Sine Function Formulas

Sine function formulas, which include both addition and subtraction formulas, are tools allowing us to compute the trigonometric values of angles that are otherwise challenging to calculate directly. In the given exercise, after applying these formulas and discovering repeating terms, simplification becomes straightforward.
Starting from the identity \( \sin(x+\frac{\pi}{3})+\sin(x-\frac{\pi}{3})=\sin x \), the simplification was driven by recognizing how:

• \( \sin x \cos \frac{\pi}{3} \) reappeared
• \( \cos \frac{\pi}{3} = \frac{1}{2} \)

These observations allowed further reduction to support the identity. Understanding these basic formulas empower you to tackle advanced identities by breaking them into steps involving known quantities, leading you to verify complex equations with ease and accuracy.