Problem 81 - Algebra and Trigonometry

Question

Prove the identity $$\frac{\sin x+\sin 2 x+\sin 3 x+\sin 4 x+\sin 5 x}{\cos x+\cos 2 x+\cos 3 x+\cos 4 x+\cos 5 x}=\tan 3 x$$

Step-by-Step Solution

Answer

The given identity simplifies to \( \tan 3x \).

Step 1: Identify the Given Expression

The given expression is \[ \frac{\sin x + \sin 2x + \sin 3x + \sin 4x + \sin 5x}{\cos x + \cos 2x + \cos 3x + \cos 4x + \cos 5x} \] We need to prove that this simplifies to \(\tan 3x\).

Step 2: Apply Sum-to-Product Identities

Use the sum-to-product identities for sine and cosine:- \( \sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \)- \( \cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \)Group terms to use these identities on sums of sine and cosine pairs within the numerator and denominator. This method will greatly simplify the calculation.

Step 3: Simplify the Numerator

Apply the sum-to-product identity on the sine terms. Group \( \sin x + \sin 5x \) and \( \sin 2x + \sin 4x \):- \( \sin x + \sin 5x = 2 \sin 3x \cos 2x \)- \( \sin 2x + \sin 4x = 2 \sin 3x \cos x \)The middle term is \( \sin 3x \).Combine the results:- Total: \( 2 \sin 3x ( \cos 2x + \cos x ) + \sin 3x \).

Step 4: Simplify the Denominator

Now, similarly apply the identity for the cosine terms:- \( \cos x + \cos 5x = 2 \cos 3x \cos 2x \)- \( \cos 2x + \cos 4x = 2 \cos 3x \cos x \)Including the middle term \( \cos 3x \), the entire denominator simplifies to:- Total: \( 2 \cos 3x ( \cos 2x + \cos x ) + \cos 3x \).

Step 5: Combine Results and Simplify Fraction

Notice both the simplified numerator and denominator are multiples of either \( \sin 3x \) or \( \cos 3x \):\[ \frac{2 \sin 3x ( \cos 2x + \cos x ) + \sin 3x}{2 \cos 3x ( \cos 2x + \cos x ) + \cos 3x} \]Factor out \( \sin 3x \) in the numerator and \( \cos 3x \) in the denominator:- Numerator: \( \sin 3x ( 2 ( \cos 2x + \cos x ) + 1 ) \)- Denominator: \( \cos 3x ( 2 ( \cos 2x + \cos x ) + 1 ) \)The terms \( 2 ( \cos 2x + \cos x ) + 1 \) cancel each other.

Step 6: Arrive at the Identity

After cancellation, the expression simplifies to:\[ \frac{\sin 3x}{\cos 3x} = \tan 3x \]Thus, the original expression simplifies to \( \tan 3x \), proving the given identity.

Key Concepts

Sum-to-Product Identities

The Sum-to-Product identities are incredibly useful in trigonometry for simplifying complex expressions involving sine and cosine functions. They help transform the sum or difference of trigonometric functions into a product, making the equations easier to manage.

Consider the identities:

• For sine: \( \sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \)
• For cosine: \( \cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \)

These equations work by averaging the angles \(A\) and \(B\). This consolidation into factors of sine and cosine can significantly reduce the complexity of larger trigonometric expressions.

When tackling problems, these identities help break down daunting sums into manageable parts, much like breaking a large problem into smaller components.

Sine and Cosine Functions

The sine and cosine functions are foundational in trigonometry. They represent ratios derived from right-angled triangles or points on the unit circle.

**Sine Function (\(\sin\))**

• Sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse.
• In the unit circle, \(\sin(\theta)\) accounts for the y-coordinate of a point.

**Cosine Function (\(\cos\))**

• Cosine is the ratio of the adjacent side to the hypotenuse in a right triangle.
• In the unit circle, \(\cos(\theta)\) measures the x-coordinate of a point.

When dealing with expressions like \( \sin x + \sin 5x \) and \( \cos x + \cos 5x \), these functions can often be paired and simplified using trigonometric identities to find solutions more easily.

Trigonometric Simplification

Trigonometric simplification is the process of transforming complex trigonometric expressions into simpler or more useful forms. This is often essential for solving equations or proving identities.

**Why Simplify?**

• Simplification makes expressions easier to understand and compare.
• It allows us to solve trigonometric equations efficiently.
• Simplifying beforehand can assist in finding common factors for cancellation or further manipulations.

In this example, we took complex sums of trigonometric functions and used identities to manipulate these into simpler terms. By employing the Sum-to-Product identities, we broke down the given terms into factors of \(\sin 3x\) and \(\cos 3x\).

Finally, when the numerator and denominator were structured similarly, a simple cancellation revealed the identity \(\tan 3x\). Such simplification steps make proofs and solutions not only accurate but elegant.