Problem 72 - Algebra and Trigonometry

Question

Solve the equation by first using a sum-to-product formula. $$\sin 5 x-\sin 3 x=\cos 4 x$$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{\pi}{6} + 2k\pi \) and \( x = \frac{5\pi}{6} + 2k\pi \).

Step 1: Apply sum-to-product formula

Use the sum-to-product identity for sine: \( \sin A - \sin B = 2\cos\left(\frac{A + B}{2}\right)\sin\left(\frac{A - B}{2}\right) \). Here, set \( A = 5x \) and \( B = 3x \), thus \( \sin 5x - \sin 3x = 2\cos(4x)\sin(x) \).

Step 2: Substitute and simplify equation

Substitute the expression from the identity into the equation: \( 2\cos(4x)\sin(x) = \cos(4x) \). Now simplify: divide both sides by \( \cos(4x) \), yielding \( 2\sin(x) = 1 \).

Step 3: Solve for \( x \)

Solve \( 2\sin(x) = 1 \) for \( x \). Divide by 2 to get \( \sin(x) = \frac{1}{2} \). The solutions for \( \sin(x) = \frac{1}{2} \) are \( x = \frac{\pi}{6} + 2k\pi \) and \( x = \frac{5\pi}{6} + 2k\pi \) for any integer \( k \).

Key Concepts

Sum-to-Product Formulas

The sum-to-product formulas are useful tools in trigonometry that help simplify expressions and solve equations involving trigonometric functions. In essence, these formulas transform sums or differences of sines and cosines into products. This change in form can make seemingly complex problems much easier to handle.

In our given problem, we encounter a difference of sines: \(\sin 5x - \sin 3x\). By applying the sum-to-product identity \(\sin A - \sin B = 2\cos\left(\frac{A + B}{2}\right)\sin\left(\frac{A - B}{2}\right)\), we convert the left side of the equation into a product term. This identity allows us to simplify the calculation considerably.

Remember, the main advantage here is transforming addition or subtraction into multiplication, which often simplifies the solving process. Knowing and applying these formulas is a key skill in working through complex trigonometric equations.

Sine Function

The sine function, one of the fundamental trigonometric functions, is often abbreviated as "sin." It has a characteristic wave shape and is periodic with a period of \(2\pi\).

In trigonometry, the sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the hypotenuse. However, in terms of the unit circle, it is the y-coordinate of a point on the circle corresponding to a given angle from the positive x-axis.

For the equation at hand, we reach a point where we solve \(2\sin(x) = 1\). This leads to \(\sin(x) = \frac{1}{2}\). To solve this, it's important to know the standard angles where the sine function yields \(\frac{1}{2}\).

• \(x = \frac{\pi}{6}\)
• \(x = \frac{5\pi}{6}\)

These solutions consider the periodic nature of sine, accounting for the fact that we repeat every full cycle of \(2\pi\). Therefore, the full solution is \(x = \frac{\pi}{6} + 2k\pi\) and \(x = \frac{5\pi}{6} + 2k\pi\), where \(k\) is any integer.

Cosine Function

Cosine, abbreviated as "cos," is another primary trigonometric function closely linked to the sine function. It is also periodic with a period of \(2\pi\) and typically represents the x-coordinate of a point on the unit circle corresponding to a specific angle from the positive x-axis.

In the context of our exercise, the cosine function appears in the transformation of the original equation through the sum-to-product formula. In the sum-to-product identity used here, \(\cos\left(\frac{A + B}{2}\right)\) emerges, showing how cosine dynamically interacts with sine.

Understanding the cosine function is crucial when simplifying trigonometric expressions. It also features prominently in verifying solutions, as simplifying terms often involves balancing sine and cosine values. In this exercise, dividing each side by \(\cos(4x)\) is a key simplification move, allowing us to focus solely on the sine function to resolve the equation.