L_83_Yash Ghoderao

2024-03-12 01:36:23

Utilize Trigonometric Identities:We are given tan(2A) = cot(A - 18°).We know: cot(x) = 1 / tan(x).Substitute and Simplify:Substitute the identity: tan(2A) = 1 / tan(A - 18°).Flip both sides: tan(A - 18°) = 1 / tan(2A).Double Angle Identity:Use the double angle identity for tangent: tan(2x) = (2 tan(x)) / (1 - tan²(x)).Substitute and Solve:Substitute the double angle identity: tan(A - 18°) = 1 / [(2 tan(A)) / (1 - tan²(A))].Multiply both sides by the denominator on the right: tan(A - 18°) (1 - tan²(A)) = 1.Matching Angles:The equation suggests that tan(A - 18°) and 1 / tan(A) have the same value.Looking at the tangent table or considering the unit circle, these conditions are met when:A - 18° = A (meaning 18° = 0°, which isn't true) ORA - 18° = 180° - A (complementary angles have the same tangent value)Solve for A:Consider the second solution (A - 18° = 180° - A):Add A to both sides: 2A = 198°.Since 2A is given as an acute angle (less than 90°), we can divide both sides by 2: A = 99°.Verify Constraint:We found A = 99°.Since 2 99° (198°) is indeed greater than 90°, this solution satisfies the constraint of 2A being acute.Therefore, the value of A in this scenario is A = 99°.