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.
ScanSolve AI Answer
Step 1: Identify the given information and define the sides of a right-angled triangle. Given Tan A = (3)/(8). In a right-angled triangle, Tan A = OppositeAdjacent. Let the Opposite side be 3x and the Adjacent side be 8x for some constant x. Step 2: Calculate the Hypotenuse using the Pythagorean theorem. Hypotenuse^2 = Opposite^2 + Adjacent^2 Hypotenuse^2 = (3x)^2 + (8x)^2 Hypotenuse^2 = 9x^2 + 64x^2 Hypotenuse^2 = 73x^2 Hypotenuse = sqrt(73x^2) = xsqrt(73) Step 3: Calculate Sin A. Sin A = OppositeHypotenuse = (3x)/(xsqrt(73)) = (3)/(sqrt(73)) To rationalize the denominator: Sin A = (3)/(sqrt(73)) × sqrt(73)sqrt(73) = 3sqrt(73)73 The value of Sin A is 3sqrt(73)73. Step 4: Calculate Cos A. Cos A = AdjacentHypotenuse = (8x)/(xsqrt(73)) = (8)/(sqrt(73)) To rationalize the denominator: Cos A = (8)/(sqrt(73)) × sqrt(73)sqrt(73) = 8sqrt(73)73 The value of Cos A is 8sqrt(73)73. Step 5: Calculate Sec A. Sec A is the reciprocal of Cos A. Sec A = (1)/(Cos A) = (1)/(8)sqrt(73) = sqrt(73)8 The value of Sec A is sqrt(73)8. Step 6: Calculate Cosec A. Cosec A is the reciprocal of Sin A. Cosec A = (1)/(Sin A) = (1)/(3)sqrt(73) = sqrt(73)3 The value of Cosec A is sqrt(73)3. Step 7: Calculate Cot A. Cot A is the reciprocal of Tan A. Cot A = (1)/(Tan A) = (1)/(3)8 = (8)/(3) The value of Cot A is (8)/(3).