If 3 sin θ + 5 cos θ = 5, then |5 sin θ - 3 cos θ| = ?

Appeared in: AIIMS BHOPAL NO- 2018( Shift-2nd)

Explanation

The problem is solved by squaring both the given equation and the target expression.
Adding the two squared equations simplifies the expression by canceling out the sin θ cos θ terms.
The fundamental trigonometric identity sin²θ + cos²θ = 1 is used to reduce the equation.
Solving the resulting algebraic equation for the unknown variable gives x² = 9.
The absolute value of the expression is therefore √9, which is 3.

Option A: The value 5 is the constant on the right side of the given equation. It is a common mistake to assume the result will be the same value without performing the correct calculation.
Option B: This is a duplicate of option A and is incorrect for the same reason. The answer is not simply the constant from the original equation.
Option D: This is also a duplicate of option A. The calculation √(3² + 5² - 5²) = √9 = 3 shows that 5 is not the correct answer.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Solving trigonometric equations using algebraic manipulation and identities as background academic context rather than a clinical decision trigger.
This is a question from the non-nursing subject of Mathematics (Trigonometry).
While trigonometry itself has applications in fields like medical imaging (CT scans, MRI) and biomechanics, this specific type of identity problem is primarily an exercise in algebraic manipulation and does not have a direct, routine application in bedside nursing practice.
What if? - If the stem changed one defining clue so that 3 no longer matched, reassess the option whose mechanism, classification, or indication now fits the revised presentation.

How to Approach the Question

Recognize the question format: a sin θ + b cos θ = c, and you need to find b sin θ - a cos θ.
Let the unknown expression be x.
Square the given equation and the equation with x.
Add the two resulting squared equations together.
Look for opportunities to simplify using the identity sin²θ + cos²θ = 1.
Solve the final algebraic equation for x and take the absolute value as required.

Concept Tested & Keywords

Concept Tested: Solving trigonometric equations using algebraic manipulation and identities.
Stem keywords: sin θ, cos θ, trigonometry
Lead-in keywords: If, then, value of

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Practise the full AIIMS BHOPAL NO- 2018( Shift-2nd)

Attempt every question from this paper in a timed mock, then review the full solution for each one.