Difficulty: Easy
Correct Answer: 2√2
Explanation:
Introduction / Context:
This question checks understanding of standard trigonometric values and the relationship between sine and cosine for special angles. When sinθ equals cosθ for an acute angle, θ must be a familiar standard angle, and then we can evaluate secθ and cosecθ directly.
Given Data / Assumptions:
Concept / Approach:
For acute angles, sinθ = cosθ occurs at θ = 45°. Using standard trigonometric values for 45°, we can compute sinθ, cosθ, secθ, and cosecθ, and then add secθ and cosecθ. No complicated identities are needed beyond definitions.
Step-by-Step Solution:
Step 1: Recall that sinθ = cosθ for an acute angle implies θ = 45°.
Step 2: For θ = 45°, sin45° = √2/2 and cos45° = √2/2.
Step 3: secθ is the reciprocal of cosθ, so sec45° = 1 / (√2/2) = 2/√2 = √2.
Step 4: Similarly, cosecθ is the reciprocal of sinθ, so cosec45° = 1 / (√2/2) = √2.
Step 5: Therefore secθ + cosecθ = √2 + √2 = 2√2.
Verification / Alternative check:
We can also reason from sinθ = cosθ by dividing both sides by cosθ (cosθ is non zero for an acute angle). That gives tanθ = 1, which again yields θ = 45°. Using any consistent value for sin45° and cos45° confirms secθ + cosecθ = 2√2, so the result is robust.
Why Other Options Are Wrong:
Option A (2) would require both secθ and cosecθ to equal 1, which only happens at θ = 90° or θ = 0° and is not allowed here. Option B (√2) would mean one of them is zero or negative, which is impossible for an acute angle. Option D (3√2) exaggerates the sum, and Option E (4) does not correspond to standard values at any simple acute angle.
Common Pitfalls:
Students sometimes confuse sinθ = cosθ with θ = 30° or θ = 60°, which are associated with √3/2 and 1/2 values. Another mistake is to forget to take reciprocals correctly and to miscompute 1/(√2/2). Always rewrite such expressions step by step to avoid algebraic slips.
Final Answer:
The correct value of secθ + cosecθ is 2√2.
Discussion & Comments