Inverse trig equations

To solve an equation with inverse trig functions:

• Rewrite the terms with formulas from the last lesson, so that the equation has fewer inverse functions.
• Solve the ordinary equation that is left.
• Check every answer in the original equation: formulas and squaring can bring in answers that do not really work.

A useful formula: tan⁻¹x + tan⁻¹y = tan⁻¹((x + y)/(1 – xy)), for xy < 1. It comes straight from tan(A + B).

Example: tan⁻¹(2x) + tan⁻¹(3x) = π/4. Combine: tan⁻¹(5x/(1 – 6x²)) = π/4, so 5x/(1 – 6x²) = 1. That gives 6x² + 5x – 1 = 0, so x = 1/6 or x = –1.

Now check. For x = –1 the left side is tan⁻¹(–2) + tan⁻¹(–3), both negative, so it cannot equal the positive π/4. Only x = 1/6 works. (Also, at x = –1, xy = 6 is not less than 1, so the formula did not even apply.)

Changing one inverse function into another: draw a right triangle. For θ = tan⁻¹x, take the opposite side x and the adjacent side 1; the hypotenuse is √(1 + x²). Every trig ratio of θ can now be read off the triangle. For example, cos(tan⁻¹x) = 1/√(1 + x²).