Problem 74 - Algebra and Trigonometry

Question

59–76 Prove the identity. $$\frac{\sin x+\sin 3 x+\sin 5 x}{\cos x+\cos 3 x+\cos 5 x}=\tan 3 x$$

Step-by-Step Solution

Answer

The identity holds by simplifying both numerator and denominator to terms involving \( \\sin 3x \) and \( \\cos 3x \).

Step 1: Recall Basic Trigonometric Identities

We'll use the trigonometric identity for the tangent of an angle, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Similarly, recall that the tangent of a sum of angles can sometimes be simplified using transformations.

Step 2: Simplify Sine Terms

Use the sum-to-product identities to simplify \( \sin x + \sin 3x + \sin 5x \). The identity \( \sin A + \sin B = 2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right) \) can be applied iteratively.

Step 3: Simplify Cosine Terms

Likewise, apply the sum-to-product identities to simplify \( \cos x + \cos 3x + \cos 5x \). Use \( \cos A + \cos B = 2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right) \) iteratively after calculating reasonable grouping.

Step 4: Use Sum-to-Product Identities

For \( \sin x + \sin 5x \) and \( \cos x + \cos 5x \), use the sum-to-product identities. Both simplify aspects in terms of \( \sin 3x \) and \( \cos 3x \).

Step 5: Analyze the Expression

The expression reduces with common factors, simplifying to \( \frac{\text{term involving } \sin 3x}{\text{term involving } \cos 3x} \). This form suggests \( \tan 3x = \frac{\sin 3x}{\cos 3x} \).

Step 6: Prove Equality

Finally, simplify the left side using algebraic manipulations until it matches \( \tan 3x \), demonstrating the original statement's validity.

Key Concepts

Sum-to-Product Identities

Sum-to-product identities are a set of trigonometric formulas that allow us to transform the sum or difference of sine or cosine functions into a product of trigonometric functions. This transformation is incredibly useful in simplifying trigonometric expressions, making them easier to work with. When you encounter expressions like \( \sin A + \sin B \), you can use the identity:

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

And for cosine sums, the identity is:

• \( \cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right) \)

These identities help to reduce the complexity of the expression. After simplifying each part, we continue exploring other elements like further simplifications or direct comparisons. They are essential when proving identities or solving trigonometric equations because they break down complex terms into more manageable pieces.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves breaking down complex trigonometric functions into simpler forms. This process often uses identities like the sum-to-product identities or even basic trigonometric equations to find equivalencies or simpler forms.

• We often start with basic identities, such as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), to see if any parts can be rewritten.
• Next, we apply suitable trigonometric identities such as the sum-to-product to rewrite expressions like \( \sin x + \sin 3x + \sin 5x \).
• Simplification might also involve factoring or recognizing patterns like perfect squares or cubes in the expressions.

After applying these techniques, the expression typically becomes straightforward, revealing any hidden relationships or simple fractions that weren’t initially apparent. The goal is to express everything in terms of simpler functions or specific known values to draw conclusions or prove identities.

Tangent of an Angle

The tangent of an angle, represented as \( \tan \theta \), fundamentally represents the ratio of the sine of the angle to the cosine of the angle. When given an expression, simplifying it to a form \( \frac{\sin \theta}{\cos \theta} \) aligns it with \( \tan \theta \), which simplifies comparison or proof tasks.

• The trigonometric identity tells us that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), which is crucial in simplifying to show tan forms.
• In our exercise, the objective was to show equivalence to \( \tan 3x \), leading to assessments like simplifying the given expression \( \frac{\sin x+\sin 3x+\sin 5x}{\cos x+\cos 3x+\cos 5x} \) through algebraic or identity-based means.
• The simplified form should ideally match \( \frac{\sin 3x}{\cos 3x} \) establishing \( \tan 3x \) through equality.

By understanding \( \tan \theta \) through this lens, we approach trigonometric problems systematically, ensuring each part logically leads to the desired result. This is foundational for tackling more complex expressions and equations.