Solving arctan(x) + arctan(2x) = π/4

Trigonometry · Axiom Academy

EXAMPLE Solving arctan(x) + arctan(2x) = π/4 Master the tangent addition formula and algebraic techniques to solve complex inverse trig equations. Excellent work! You've successfully solved a complex inverse trigonometric equation. Here's what we learned: Strategic Approach: When dealing with sums of inverse trig functions, taking the tangent (or appropriate trig function) of both sides often simplifies the problem. Tangent Addition Formula: The formula tan(A + B) = (tan A + tan B)/(1 - tan A · tan B) is essential for solving these equations. Inverse Function Property: Remember that tan(arctan(x)) = x, which allows us to simplify complex expressions. Algebraic Transformation: Converting the trigonometric equation into a quadratic equation makes it solvable using familiar algebraic techniques. Domain Verification: Always check that solutions are valid - especially with inverse trig functions, certain values may be extraneous or outside the domain. Exact vs. Approximate: The answer x = (-3 + √17)/4 is exact. While it's approximately 0.281, keeping it in exact form is mathematically precise. This problem demonstrates the power of combining trigonometric identities with algebraic techniques. Similar approaches work for equations involving arcsin, arccos, and other inverse functions!

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