Difficulty: Easy
Correct Answer: √2
Explanation:
Introduction / Context:This question combines basic trigonometric values at standard angles with simple algebraic manipulation. You are asked to evaluate an expression involving sine and cotangent for an angle of 45°, which is a common special angle in right triangles.
Given Data / Assumptions:
Concept / Approach:Use the standard trigonometric values for 45°. In a right triangle with one acute angle of 45°, the other acute angle is also 45°, making it an isosceles right triangle. The key values are Sin 45° = √2 / 2, Cos 45° = √2 / 2 and Tan 45° = 1. Cotangent is the reciprocal of tangent, so Cot 45° = 1. Substitute these values into the expression and simplify.
Step-by-Step Solution:Step 1: Identify the angle in the expression: F = 45°.Step 2: Recall the standard values: Sin 45° = √2 / 2 and Cot 45° = 1.Step 3: Substitute into the expression 2 Sin F × Cot F: 2 * Sin 45° * Cot 45° = 2 * (√2 / 2) * 1.Step 4: Simplify: 2 * (√2 / 2) = √2.Step 5: The value of the whole expression is therefore √2.
Verification / Alternative check:You can also express Cot 45° as Cos 45° / Sin 45°. Then the expression becomes 2 * Sin 45° * (Cos 45° / Sin 45°) = 2 * Cos 45°. Since Cos 45° = √2 / 2, the result is 2 * (√2 / 2) = √2, which confirms the previous calculation.
Why Other Options Are Wrong:The values 2, 1/√2, 1/2 and √3/2 correspond to other combinations of trig functions or angles and do not match the calculation of 2 Sin 45° × Cot 45°. None of these equals √2 when evaluated.
Common Pitfalls:Some learners misremember the standard trigonometric values or confuse cotangent with cosecant. Others may incorrectly square the factor 2 or forget to simplify the product fully. Keeping the key 45° values in mind and substituting carefully prevents these mistakes.
Final Answer:The value of 2 Sin F × Cot F when angle F is 45° is √2.
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