Solving the Trigonometric Equation

We are given a trigonometric equation and asked to find the value of a related expression. The given equation is:

\(3 \sin \theta + 5 \cos \theta = 5 \quad \quad (1)\)

We need to find the value of the expression:

\(5 \sin \theta - 3 \cos \theta\)

Let's denote the value we need to find as \(x\).

\(5 \sin \theta - 3 \cos \theta = x \quad \quad (2)\)

Algebraic Manipulation Strategy

A common technique to solve problems of this type involves squaring both equations and combining them. This utilizes the fundamental trigonometric identity \(\sin^2 \theta + \cos^2 \theta = 1\).

Step 1: Square Equation (1)

Squaring both sides of equation (1):

\((3 \sin \theta + 5 \cos \theta)^2 = 5^2\)

Expanding the left side using \((a+b)^2 = a^2 + 2ab + b^2\):

\( (3 \sin \theta)^2 + 2(3 \sin \theta)(5 \cos \theta) + (5 \cos \theta)^2 = 25 \)

\( 9 \sin^2 \theta + 30 \sin \theta \cos \theta + 25 \cos^2 \theta = 25 \quad \quad (3) \)

Step 2: Square Equation (2)

Squaring both sides of equation (2):

\((5 \sin \theta - 3 \cos \theta)^2 = x^2\)

Expanding the left side using \((a-b)^2 = a^2 - 2ab + b^2\):

\( (5 \sin \theta)^2 - 2(5 \sin \theta)(3 \cos \theta) + (3 \cos \theta)^2 = x^2 \)

\( 25 \sin^2 \theta - 30 \sin \theta \cos \theta + 9 \cos^2 \theta = x^2 \quad \quad (4) \)

Step 3: Combine the Squared Equations

Now, let's add equation (3) and equation (4):

\((9 \sin^2 \theta + 30 \sin \theta \cos \theta + 25 \cos^2 \theta) + (25 \sin^2 \theta - 30 \sin \theta \cos \theta + 9 \cos^2 \theta) = 25 + x^2\)

Notice that the cross terms (\(30 \sin \theta \cos \theta\) and \(-30 \sin \theta \cos \theta\)) cancel each other out:

\( 9 \sin^2 \theta + 25 \sin^2 \theta + 25 \cos^2 \theta + 9 \cos^2 \theta = 25 + x^2 \)

Group the \(\sin^2 \theta\) and \(\cos^2 \theta\) terms:

\( (9+25) \sin^2 \theta + (25+9) \cos^2 \theta = 25 + x^2 \)

\( 34 \sin^2 \theta + 34 \cos^2 \theta = 25 + x^2 \)

Step 4: Apply the Pythagorean Identity

Factor out 34 and use the identity \(\sin^2 \theta + \cos^2 \theta = 1\):

\( 34 (\sin^2 \theta + \cos^2 \theta) = 25 + x^2 \)

\( 34 (1) = 25 + x^2 \)

\( 34 = 25 + x^2 \)

Step 5: Solve for \(x\)

Rearrange the equation to solve for \(x^2\):

\( x^2 = 34 - 25 \)

\( x^2 = 9 \)

Taking the square root of both sides gives:

\( x = \pm \sqrt{9} \)

\( x = \pm 3 \)

Determining the Specific Value

We have two possible values for \(x\): \(3\) and \(-3\). To find the specific value, let's examine the original equation \(3 \sin \theta + 5 \cos \theta = 5\).

Consider the case where \(\cos \theta = 1\). For this to be possible, \(\sin \theta\) must be \(0\) (since \(\sin^2 \theta + \cos^2 \theta = 1\)).

Let's check if \(\sin \theta = 0\) and \(\cos \theta = 1\) satisfy the given equation:

\( 3(0) + 5(1) = 0 + 5 = 5 \)

This pair of values (\(\sin \theta = 0, \cos \theta = 1\)) satisfies the given condition.

Now, substitute these values into the expression we want to find (\(5 \sin \theta - 3 \cos \theta\)):

\( 5(0) - 3(1) = 0 - 3 = -3 \)

Therefore, the value of \(5 \sin \theta - 3 \cos \theta\) is \(-3\). This matches one of the possible values we found for \(x\).

Final Answer Summary

The steps involved using algebraic manipulation and the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to find possible values. By testing a specific valid case (\(\sin \theta = 0, \cos \theta = 1\)), we determined the unique value of the expression \(5 \sin \theta - 3 \cos \theta\) to be \(-3\).