Problem 85 - Algebra and Trigonometry

Question

\(73-90\) Prove the identity. $$ \frac{\sin x+\sin 5 x}{\cos x+\cos 5 x}=\tan 3 x $$

Step-by-Step Solution

Answer

The identity is proven as \( \tan 3x = \frac{\sin 3x}{\cos 3x} \).

Step 1: Identify Known Identities

The exercise involves simplifying the expression \( \frac{\sin x+\sin 5 x}{\cos x+\cos 5 x} \) to prove that it equals \( \tan 3x \). To begin simplifying, note that this could involve using sum-to-product identities for sine and cosine.

Step 2: Apply Sum-to-Product Identities

Use the sum-to-product identities: \( \sin a + \sin b = 2 \sin\left(\frac{a+b}{2}\right) \cos\left(\frac{a-b}{2}\right) \) and \( \cos a + \cos b = 2 \cos\left(\frac{a+b}{2}\right) \cos\left(\frac{a-b}{2}\right) \). Applying them gives:1. \( \sin x + \sin 5x = 2 \sin\left(3x\right) \cos\left(2x\right) \)2. \( \cos x + \cos 5x = 2 \cos\left(3x\right) \cos\left(2x\right) \).

Step 3: Simplify the Expression

Substitute the results of step 2 back into the expression:\[\frac{2 \sin(3x) \cos(2x)}{2 \cos(3x) \cos(2x)}\]This simplifies to \( \frac{\sin(3x)}{\cos(3x)} \) because the \( 2 \cos(2x) \) terms cancel out.

Step 4: Recognize Trigonometric Identity

Note that \( \frac{\sin(3x)}{\cos(3x)} \) is equal to \( \tan(3x) \). Therefore, the original expression simplifies correctly to prove the identity.

Key Concepts

Sum-to-Product Identities

Sum-to-product identities are powerful tools in trigonometry that allow us to convert sums of trigonometric functions into products. These identities are particularly useful when simplifying expressions or proving equations involving trigonometric terms. Let's explore the basic forms of these identities and how they work.

Here are two key sum-to-product identities:

• The identity for sine: \(\sin a + \sin b = 2 \sin\left(\frac{a+b}{2}\right) \cos\left(\frac{a-b}{2}\right) \)
• The identity for cosine: \(\cos a + \cos b = 2 \cos\left(\frac{a+b}{2}\right) \cos\left(\frac{a-b}{2}\right) \)

Using these identities simplifies the process of working with more complex trigonometric expressions. For instance, in our given expression, by applying the sine identity to \(\sin x + \sin 5x\), we simplify it to \(2 \sin(3x) \cos(2x)\), and similarly for the cosine terms. This clever transformation is what makes these identities incredibly useful in trigonometric simplification.

Sine and Cosine Functions

Sine and cosine functions are fundamental trigonometric functions that are essential in the study of mathematics, especially in trigonometry and calculus. They describe the behavior of periodic phenomena, such as waves.

Main Characteristics:

• The sine function, \(\sin x\), represents the y-coordinate of a point on the unit circle at an angle \(x\).
• The cosine function, \(\cos x\), represents the x-coordinate of a point on the unit circle at the same angle.

Their periodic nature means that they repeat their values in regular intervals. Both functions have a period of \(2\pi\), meaning they repeat their values every \(2\pi\) radians.

In our exercise, the sum-to-product identities utilize both the sine and cosine functions. By converting sums of these functions into products, we make it easier to manipulate and simplify expressions, particularly when they occur as part of more complex equations like those found in trigonometric identities.

Trigonometric Simplification

Trigonometric simplification involves reducing more complex trigonometric expressions into simpler or more manageable forms. This process often uses known identities and properties of trigonometric functions.

In our example problem, trigonometric simplification is achieved through several steps:

• First, recognize when to apply sum-to-product identities to manage terms like \(\sin x + \sin 5x\) and \(\cos x + \cos 5x\).
• By applying these identities, we transform the expression and can later cancel terms common to both the numerator and denominator.
• The result is then in a simplified form \(\frac{\sin(3x)}{\cos(3x)}\), which can be easily converted to \(\tan(3x)\), hence proving the given identity.

This process not only confirms the identity but also strengthens our understanding of how various trigonometric functions interact. Mastering trigonometric simplification allows for solving complex equations more efficiently, a skill enhanced by knowing the right identities to apply.