Problem 18 - Algebra and Trigonometry with Analytic Geometry

Question

Verify the identity. $$ \frac{\sin \theta+\sin 3 \theta}{\cos \theta+\cos 3 \theta}=\tan 2 \theta $$

Step-by-Step Solution

Answer

The identity is verified: \( \frac{\sin \theta + \sin 3\theta}{\cos \theta + \cos 3\theta} = \tan 2\theta \).

Step 1: Express sine and cosine terms using angle addition formulas

Start by using the angle addition formulas for the sine and cosine in the numerator and denominator. We know that \( \sin(3\theta) = 3\sin\theta - 4\sin^3\theta \) and \( \cos(3\theta) = 4\cos^3\theta - 3\cos\theta \). Substitute these into the expression.

Step 2: Simplify the expression in the numerator

Substitute the expanded forms: \( \sin \theta + \sin 3\theta = \sin \theta + 3\sin \theta - 4\sin^3\theta = 4\sin\theta - 4\sin^3\theta \). Factor out \( 4\sin \theta \) to obtain: \( 4\sin \theta(1 - \sin^2 \theta) \). Using the identity \( 1 - \sin^2\theta = \cos^2\theta \), this becomes \( 4\sin\theta\cos^2\theta \).

Step 3: Simplify the expression in the denominator

Substitute the expanded forms: \( \cos \theta + \cos 3\theta = \cos \theta + 4\cos^3\theta - 3\cos \theta = 4\cos^3\theta - 2\cos \theta \). Factor out \( 2\cos \theta \) to obtain: \( 2\cos \theta(2\cos^2 \theta - 1) \). Using the identity \( 2\cos^2 \theta - 1 = \cos 2\theta \), this becomes \( 2\cos \theta\cos 2\theta \).

Step 4: Simplify the fraction

The fraction \( \frac{4\sin\theta\cos^2\theta}{2\cos\theta\cos 2\theta} \) simplifies by cancelling out \( 2\cos\theta \) from both the numerator and denominator. This leaves \( \frac{2\sin\theta\cos\theta}{\cos 2\theta} \).

Step 5: Use the double angle identity for sine

Recall the double angle formula for sine: \( \sin 2\theta = 2\sin \theta \cos \theta \). Substitute this into the expression to get \( \frac{\sin 2\theta}{\cos 2\theta} \).

Step 6: Identify tan 2θ

Recognize that \( \frac{\sin 2\theta}{\cos 2\theta} \) is exactly \( \tan 2\theta \). Therefore, the original identity is verified: \( \frac{\sin \theta + \sin 3\theta}{\cos \theta + \cos 3\theta} = \tan 2\theta \).

Key Concepts

Angle Addition Formulas

The angle addition formulas are essential tools in trigonometry used to express the trigonometric functions of sums of angles in terms of the functions of each angle separately. For example, the formulas involve calculations like

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

These expressions help in breaking down complex angle functions into simpler parts that can be easier to work with. In the given exercise, using the triple angle formulas for sine and cosine allowed the breakdown into manageable trigonometric components that further simplified the process of verifying the identity.

Double Angle Formulas

Double angle formulas are special cases of the angle addition formulas, where both angles are the same. They are instrumental in simplifying expressions and solving equations involving trigonometric functions. The most common ones include:

• \( \sin 2\theta = 2\sin \theta \cos \theta \)
• \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)
• Another form \( \cos 2\theta = 2\cos^2 \theta - 1 \)
• And another form \( \cos 2\theta = 1 - 2\sin^2 \theta \)

In the solution, the double angle formula \( \sin 2\theta = 2\sin \theta \cos \theta \) was pivotal for expressing the result in terms of \( \tan 2\theta \), thus verifying the identity.

Factoring in Trigonometry

Factoring is a vital skill in algebra and trigonometry that helps in simplifying expressions and solving equations. By factoring trigonometric equations, complex expressions can be reduced to more straightforward ones that are easier to solve or verify. In our exercise:

• \( \sin \theta + 3\sin\theta - 4\sin^3\theta = 4\sin \theta(1 - \sin^2 \theta) \)
• Using the identity \( 1 - \sin^2 \theta = \cos^2 \theta \), this further simplifies to \( 4\sin \theta \cos^2 \theta \)
• Similarly, by factoring \( \cos \theta \) in the denominator and applying trigonometric identities, the expression was streamlined for easier completion.

This process shows that familiarity with factoring techniques and trigonometric identities significantly aids in simplifying and solving such exercises effectively.

Verifying Identities

Verifying trigonometric identities requires a combination of algebraic manipulation and the application of various trigonometric formulas and identities. The goal is to show that two seemingly different expressions are, in fact, equal. To verify an identity effectively:

• Start by simplifying both sides of the equation independently.
• Use identities such as angle addition, double angle formulas, and others to transform the sides.
• Simplify step-by-step, factor when necessary, and cancel terms appropriately.

In our example, the goal was to demonstrate that \( \frac{\sin \theta + \sin 3\theta}{\cos \theta + \cos 3\theta} \) simplifies to \( \tan 2\theta \). This required employing multiple identities and simplifying until reaching the desired form. Patience and meticulous attention to detail are key in verifying identities successfully.