This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the questions: Question 40: If = k, find . Step 1: Use the Pythagorean identity to find . We know that ^2 + ^2 = 1. Substituting = k: k^2 + ^2 = 1 ^2 = 1 - k^2 = sqrt(1 - k^2) Step 2: Use the definition of . = ( )/( ) Substitute the expressions for and : = (k)/(sqrt(1 - k^2)) The correct option is A. Question 41: Evaluate 101.5^2 - 100.5^2. Step 1: Apply the difference of squares formula, a^2 - b^2 = (a - b)(a + b). Here, a = 101.5 and b = 100.5. 101.5^2 - 100.5^2 = (101.5 - 100.5)(101.5 + 100.5) Step 2: Perform the arithmetic. (101.5 - 100.5) = 1 (101.5 + 10 ✂️ _That answer was long and got cut off. Reply continue and I'll finish it._