Difficulty: Medium
Correct Answer: (2√2 - √3)/2
Explanation:
Introduction / Context:
This trigonometry problem combines knowledge of right triangles, special angles, and cosecant. You are given a right triangle with one acute angle specified and asked to compute an expression involving cosec P and a surd term. Understanding how the angles in a right triangle relate and recalling exact trigonometric values for special angles are essential here.
Given Data / Assumptions:
Concept / Approach:
In a right triangle, the two acute angles are complementary. If angle R is 45°, then the other acute angle P must also be 45°. For angle P = 45°, sin P has a standard value, and cosec P is its reciprocal. Once we know cosec P, we subtract √3 / 2 and simplify the resulting surd expression to match one of the given answer options.
Step-by-Step Solution:
Verification / Alternative check:
You can approximate the values numerically. Take √2 ≈ 1.414 and √3 ≈ 1.732. Then cosec P − √3 / 2 ≈ 1.414 − 0.866 = 0.548. Now evaluate (2√2 − √3) / 2 ≈ (2 * 1.414 − 1.732) / 2 ≈ (2.828 − 1.732) / 2 ≈ 1.096 / 2 ≈ 0.548, which matches, confirming that the simplified expression is correct.
Why Other Options Are Wrong:
The expressions (3√3 − 1)/3, 2/√3, and (2 − √3)/√3 correspond to different numerical values and do not equal √2 − √3 / 2. The option √2 − √3/2 is equivalent in meaning but not written as a single fraction, while the correct choice in the list that matches this expression exactly in simplified fractional form is (2√2 − √3)/2.
Common Pitfalls:
Common errors include miscalculating the remaining acute angle, confusing sine with cosine, or forgetting that cosec is the reciprocal of sine. Another mistake is to fail to convert both terms to a common denominator when combining surds. Being careful with complementary angles and fraction operations is key to solving this type of question correctly.
Final Answer:
The value of cosec P − √3 / 2 is (2√2 − √3) / 2.
Discussion & Comments