Problem 18 - Precalculus: Functions and Graphs

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 as both sides are equal by using angle identities and simplification.

Step 1: Expand Trigonometric Functions

The first step is to expand the terms in the identity using trigonometric identities. Start with \(\sin 3\theta\) and \(\cos 3\theta\) using the triple angle identities:\[ \sin 3\theta = 3\sin\theta - 4\sin^3\theta \]\[ \cos 3\theta = 4\cos^3\theta - 3\cos\theta \]

Step 2: Substitute and Simplify the Numerator

Substitute \(\sin 3\theta = 3\sin\theta - 4\sin^3\theta\) into the numerator:\[ \sin \theta + \sin 3\theta = \sin \theta + 3\sin\theta - 4\sin^3\theta = 4\sin \theta - 4\sin^3 \theta \]Factor the expression:\[ = 4\sin \theta(1 - \sin^2 \theta) = 4\sin \theta \cos^2 \theta \]

Step 3: Substitute and Simplify the Denominator

Substitute \(\cos 3\theta = 4\cos^3\theta - 3\cos\theta\) into the denominator:\[ \cos \theta + \cos 3\theta = \cos \theta + 4\cos^3\theta - 3\cos\theta = 4\cos^3 \theta - 2\cos \theta \]Factor the expression:\[ = 2\cos \theta(2\cos^2 \theta - 1) = 2\cos \theta \cdot 2\cos^2 \theta - 2\cos \theta \cdot 1\]

Step 4: Simplify the Fraction

Combine the results from the simplified expressions:\[ \frac{4\sin \theta \cos^2 \theta}{2\cos \theta (2\cos^2 \theta - 1)} \]This simplifies by canceling \(2\cos \theta\):\[ = 2\tan \theta \cdot \frac{\cos^2 \theta}{2\cos^2 \theta - 1} \]

Step 5: Use Double Angle Identity for Tangent

Recall the double angle identity for tangent:\[ \tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta} \]Recognize that because of simplifications, \(\frac{\cos^2 \theta}{2\cos^2 \theta - 1}\) represents this identity, thus proving:\[ \frac{\sin \theta + \sin 3\theta}{\cos \theta + \cos 3\theta} = \tan 2\theta \]

Step 6: Verify the Steps

Review the steps to ensure accuracy in trigonometric expansions and simplifications. By following each calculation and identity, confirm that both sides of the equation match, verifying the given identity.

Key Concepts

Triple Angle Identities

Triple angle identities are part of the family of trigonometric identities used to express trigonometric functions of angles such as \(3\theta\). These identities are particularly useful when dealing with more complex expressions that contain triple angles.For example:

• \(\sin 3\theta = 3\sin\theta - 4\sin^3\theta\)
• \(\cos 3\theta = 4\cos^3\theta - 3\cos\theta\)

Understanding these expressions helps simplify trigonometric equations. In the given exercise, we used these identities to rewrite \(\sin 3\theta\) and \(\cos 3\theta\). This step allows us to transform and combine terms, paving the way to simplify the equation. Breaking down complicated trigonometric expressions using identities is a powerful technique to solve such problems.

Double Angle Identities

Double angle identities allow you to express trigonometric functions like \(2\theta\) in terms of \(\theta\). This is particularly helpful when you encounter terms like \(\tan 2\theta\), \(\sin 2\theta\), or \(\cos 2\theta\) within trigonometric equations.The identity relevant in this exercise is the double angle identity for tangent:

• \(\tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta}\)

By understanding and applying this identity, we were able to express the right side of the equation in our exercise, which helped verify that the given identity holds true. Double angle identities are crucial in solving, proving, or simplifying equations by reducing the complexity of the angle involved.

Tangent Function

The tangent function is one of the primary trigonometric functions, often symbolized as \(\tan\theta\). It represents the ratio of sine to cosine:

• \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)

The tangent function is important because it connects the other two primary functions: sine and cosine. In our exercise, we aimed to show that:

• \(\frac{\sin \theta + \sin 3\theta}{\cos \theta + \cos 3\theta} = \tan 2\theta\)

Understanding \(\tan \theta\) helps when you deal with identities involving the tangent of multiple angles, like double angle identities. Recognizing its role in trigonometric problems can simplify proving or verifying identities.

Sine Function

The sine function, denoted as \(\sin \theta\), is a fundamental trigonometric function representing the y-coordinate of a point on a unit circle. It is often involved in angles and periodic functions.In the given exercise, the sine function appears in both \(\sin \theta\) and \(\sin 3\theta\). Using the sine triple angle identity allowed us to simplify these terms to a more manageable form:

• \(\sin 3\theta = 3\sin\theta - 4\sin^3\theta\)

Through simplification, connecting sine to cosine and other trigonometric functions, we glean insights into the behavior of these functions over various intervals. Learning about the sine function's properties is key to solving and verifying complex trigonometric identities.

Cosine Function

The cosine function, noted as \(\cos \theta\), is another primary trigonometric function. It represents the x-coordinate of a point on the unit circle and is associated with the cosine law in triangles.In the exercise, \(\cos \theta\) and \(\cos 3\theta\) play crucial roles in the simplification process. We used the triple angle identity:

• \(\cos 3\theta = 4\cos^3\theta - 3\cos\theta\)

By rearranging the expression using this identity, we broke down the equation into parts that can be simplified. Understanding the behavior and properties of cosine informs how you approach problems involving angles, allowing you to apply identities efficiently. Comprehending how it works with other trigonometric functions can simplify complex expressions.