Supplemental Notes

  1. The Right Triangle Method
    Used to turn trig expressions into algebraic ones.

  • Inner function (like arctan) defines the angle.

  • Use SOH-CAH-TOA to label side 1 and side 2.

  • Use Pythagorean theorem (a^2 + b^2 = c^2) to find side 3.

  • Use outer function to write the final ratio.
    Example: cos(arccot(x)) labels Adj as x and Opp as 1. Hyp is sqrt(x^2 + 1). Result: x / sqrt(x^2 + 1).

  1. Solving by Domain
    Inverse trig equations often only have solutions at the "boundaries."

  • Arccos exists between -1 and 1.

  • Arcsec exists outside -1 and 1.

  • They only meet at x = 1 and x = -1.

  1. Graphing arcsec(6x)

  • The "6" shrinks the domain gap. The function only exists for x >= 1/6 or x <= -1/6.

  • The minimum and maximum points are the y-values at those specific x-values.

  • The horizontal asymptote is always y = pi/2 as x goes to infinity.

  1. Related Rates with Arctan
    To find the change in angle over time:

  • Step 1: Find the derivative of theta with respect to x.

  • Step 2: Multiply by the velocity (dx/dt).

  • Step 3: Simplify the algebra. If theta = arctan(5/x), the derivative dtheta/dx simplifies to -5 / (x^2 + 25).