Curioustab Logo Curioustab
Home » Aptitude » Simplification

If sin 3θ · sec 2θ = 1 for an acute angle θ, find the value of 3 · tan^2(5θ/2) − 1.

Difficulty: Hard

Correct Answer: 2

Explanation:


Introduction / Context:

This trigonometric equation problem brings together several identities and angle transformations. You are given a relation involving sin 3θ and sec 2θ, and asked to evaluate an expression with tan^2(5θ/2). This type of question is typical of higher difficulty aptitude exams and requires comfort with angle sum formulas and basic solving of trigonometric equations.



Given Data / Assumptions:

  • sin 3θ · sec 2θ = 1.
  • Angle θ is acute, so 0 degrees < θ < 90 degrees.
  • You need to find 3 · tan^2(5θ/2) − 1.
  • Standard trigonometric identities such as sec α = 1 / cos α and relationships between sine and cosine apply.


Concept / Approach:

Start by rewriting sec 2θ as 1 / cos 2θ. The given equation then becomes sin 3θ = cos 2θ. Use the identity cos α = sin(90° − α) or, in radians, cos α = sin(π/2 − α), to relate cos 2θ to a sine function. Equating two sine functions lets you find θ. Once θ is known, you can compute 5θ/2 and evaluate tan^2 of that angle, and then plug into the target expression.



Step-by-Step Solution:

Step 1: Rewrite sec 2θ as 1 / cos 2θ. Then sin 3θ · sec 2θ = 1 becomes sin 3θ / cos 2θ = 1.Step 2: Therefore sin 3θ = cos 2θ.Step 3: Use cos 2θ = sin(90° − 2θ). So sin 3θ = sin(90° − 2θ).Step 4: For acute θ, consider the principal solution sin A = sin B ⇒ A = B or A = 180° − B.Step 5: From A = B, set 3θ = 90° − 2θ. Then 5θ = 90°, so θ = 18°.Step 6: The alternative 3θ = 180° − (90° − 2θ) = 90° + 2θ gives θ = 90°, which is not acute, so discard it.Step 7: With θ = 18°, compute 5θ/2 = (5 * 18°) / 2 = 45°.Step 8: Evaluate tan 45° = 1, so tan^2(45°) = 1.Step 9: Substitute into the required expression: 3 · tan^2(5θ/2) − 1 = 3 · 1 − 1 = 2.


Verification / Alternative check:

Check that θ = 18° satisfies the original equation. Compute sin 3θ = sin 54° and cos 2θ = cos 36°. Using known exact values, sin 54° equals cos 36°, so sin 3θ / cos 2θ = 1, which matches the given equation. This confirms θ = 18° is correct and therefore the derived value 2 for the final expression is reliable.



Why Other Options Are Wrong:

The values 0, 1, 3, and 4 typically arise from using the wrong solution of the sine equation, miscalculating 5θ/2, or forgetting to square tan before multiplying by 3. Once θ is correctly found as 18°, tan^2(5θ/2) must be 1, leaving only 2 as the valid result for 3 · tan^2(5θ/2) − 1.



Common Pitfalls:

Students often confuse the relationship between sine and cosine when the angles differ, or they forget the second possible sine solution and mis-handle the acute condition. Another frequent error is mixing degrees and radians in the same calculation. Consistently treating angles in degrees and carefully tracking each transformation avoids these traps.



Final Answer:

The value of 3 · tan^2(5θ/2) − 1 is 2.


← Previous Question Next Question→

More Questions from Simplification

Discussion & Comments

No comments yet. Be the first to comment!
Join Discussion