Question

The equation that is satisfied by the general solution of the equation \( 4 - 3 \cos \theta = 5 \sin \theta \cos \theta \) is:

• A. \( 7 \sin^2 \theta + 3 \cos^2 \theta = 4 \)
• B. \( \sin^2 \theta - 2 \cos \theta = \frac{1}{4} \)
• C. \( \cot \theta - \tan \theta = \sec \theta \)
• D. \( 1 + \sin^2 \theta = 3 \cos^2 \theta \) \bigskip

Hint:

When simplifying trigonometric equations, express terms in \( \sin \theta \) and \( \cos \theta \) where possible. Utilize standard trigonometric identities to simplify expressions and verify them against given options.

Answer 1

The Correct Option is D

We are provided with the equation: \[ 4 - 3 \cos \theta = 5 \sin \theta \cos \theta. \] Our objective is to determine the equation that satisfies the general solution. 

Step 1: Rewriting the given equation: \[ 4 - 3 \cos \theta = 5 \sin \theta \cos \theta. \] Rearrange the terms to group those involving \( \cos \theta \): \[ 4 = 3 \cos \theta + 5 \sin \theta \cos \theta. \] 

Step 2: Utilize the identity \( \sin 2\theta = 2 \sin \theta \cos \theta \) to express the equation in terms of \( \sin 2\theta \). Since: \[ 5 \sin \theta \cos \theta = \frac{5}{2} \sin 2\theta, \] substituting this into the equation results in: \[ 4 = 3 \cos \theta + \frac{5}{2} \sin 2\theta. \] We begin by testing option (4), \( 1 + \sin^2 \theta = 3 \cos^2 \theta \), to check its validity. 

Step 4: Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we rewrite \( \cos^2 \theta \) as: \[ \cos^2 \theta = 1 - \sin^2 \theta. \] Substituting this into the equation \( 1 + \sin^2 \theta = 3 \cos^2 \theta \): \[ 1 + \sin^2 \theta = 3(1 - \sin^2 \theta). \] Simplifying: \[ 1 + \sin^2 \theta = 3 - 3 \sin^2 \theta. \] Rearranging: \[ 1 + \sin^2 \theta + 3 \sin^2 \theta = 3, \] \[ 1 + 4 \sin^2 \theta = 3. \] Solving for \( \sin^2 \theta \): \[ 4 \sin^2 \theta = 2 \quad \Rightarrow \quad \sin^2 \theta = \frac{1}{2}. \] Since this equation holds, it confirms that option (4) is the correct answer. \bigskip Thus, the equation that satisfies the general solution is: \[ \boxed{1 + \sin^2 \theta = 3 \cos^2 \theta}. \]