Problem 68 - Algebra and Trigonometry with Analytic Geometry

Question

Verify the identity by transforming the lefthand side into the right-hand side. $$\frac{1+\cos ^{2} 3 \theta}{\sin ^{2} 3 \theta}=2 \csc ^{2} 3 \theta-1$$

Step-by-Step Solution

Answer

Identity is verified by showing both sides simplify to \( \frac{2 - \sin^2 3\theta}{\sin^2 3\theta} \).

Step 1: Rewrite Cosecant Identity

Recall that the cosecant function is the reciprocal of the sine function: \[ \csc^2 3\theta = \frac{1}{\sin^2 3\theta} \]Therefore, \[ 2 \csc^2 3\theta = \frac{2}{\sin^2 3\theta}. \]

Step 2: Substitute and Simplify Right-Hand Side

Substitute the expression from Step 1 into the right-hand side:\[ 2 \csc^2 3\theta - 1 = \frac{2}{\sin^2 3\theta} - 1. \]Find a common denominator:\[ \frac{2}{\sin^2 3\theta} - 1 = \frac{2}{\sin^2 3\theta} - \frac{\sin^2 3\theta}{\sin^2 3\theta} = \frac{2 - \sin^2 3\theta}{\sin^2 3\theta}. \]

Step 3: Use Pythagorean Identity for Cosine

Recall the Pythagorean identity: \[ \sin^2 \theta + \cos^2 \theta = 1 \rightarrow \cos^2 \theta = 1 - \sin^2 \theta. \] Apply this identity for \( \cos^2 3\theta \):\[ \cos^2 3\theta = 1 - \sin^2 3\theta. \]

Step 4: Substitute and Simplify Left-Hand Side

Start with the left-hand side expression:\[ \frac{1 + \cos^2 3\theta}{\sin^2 3\theta}. \]Use the identity from Step 3:\[ \cos^2 3\theta = 1 - \sin^2 3\theta \rightarrow 1 + \cos^2 3\theta = 1 + (1 - \sin^2 3\theta) = 2 - \sin^2 3\theta. \]Replace in the equation:\[ \frac{2 - \sin^2 3\theta}{\sin^2 3\theta}. \]

Step 5: Verify Identity

The expression on the left-hand side is now:\[ \frac{2 - \sin^2 3\theta}{\sin^2 3\theta}, \]which matches the simplified expression obtained for the right-hand side in Step 2. This verifies the identity.

Key Concepts

Cosecant Identity

The cosecant function is an important part of trigonometry, often abbreviated as "csc." Its identity is straightforward and stems from its relationship with the sine function. Simply put, the cosecant of an angle is the reciprocal of its sine.

This means that the cosecant of any angle \(θ\) can be expressed as: \[ \csc\,θ = \frac{1}{\sin\,θ}. \]
For squared versions, we have: \[ \csc^2\,θ = \frac{1}{\sin^2\,θ}. \]
This relationship is vital in simplifying trigonometric expressions, especially when identities or transformations are involved.

In our exercise, the identity for \(\csc^2 3θ\) was used to transform part of the equation. Recognizing and applying this identity simplifies complex trigonometric identities more easily, allowing us to progress through problems with precision.

Pythagorean Identity

The Pythagorean identity is one of the cornerstone equations in trigonometry, fundamental for solving identities and equations. This identity provides a relationship between the sine and cosine functions.

It states that for any angle \(θ\): \[ \sin^2 θ + \cos^2 θ = 1. \]

From this equation, we can rearrange terms to solve for \(\cos^2 θ\)

• ew{\sin^2 θ} by subtracting \(\sin^2 θ\) from both sides, which gives: \[ \cos^2 θ = 1 - \sin^2 θ. \]

Knowing this identity helps in expressing complex trigonometric functions in terms of simpler ones, streamlining calculations in many mathematical problems.
In the exercise, this identity was utilized to express \(\cos^2 3θ\) as \(1 - \sin^2 3θ\), allowing for the transformation of the expression as needed.

Sine Function

The sine function is a fundamental concept in trigonometry. It relates to the ratio of the opposite side to the hypotenuse in a right-angled triangle.

Defined as: \[ \sin\,θ = \frac{\text{opposite}}{\text{hypotenuse}}. \]

• Measured from 0° to 360° in degrees or from 0 to \(2π\) in radians
• Repeats its values every 360°, making it a periodic function.

In our exercise, the sine function is at the center of simplifying identities. We handle expressions like \(\sin^2 3θ\) when applying identities or transformations, underscoring its significance in creating and resolving trigonometric expressions efficiently.

Cosine Function

The cosine function, much like its companion the sine function, is essential in trigonometry. It serves a similar purpose: expressing relationships in right-angled triangles. The cosine of an angle \(θ\) is the ratio of the adjacent side over the hypotenuse:

Defined as: \[ \cos\,θ = \frac{\text{adjacent}}{\text{hypotenuse}}. \]

• Also a periodic function with a period of 360° or \(2π\) radians.
• Starts at maximum value when \(θ = 0\,°\ or\ 0\) radians and repeats every full cycle.

In the exercise, you utilize the cosine function to apply the Pythagorean identity, transforming expressions to simplify the identity verification process.
This illustrates the crucial role of cosine in evaluating and balancing complex trigonometric equations.