Exam Notes on Inverse Functions and Trigonometry

Inverse Functions

  • Numerically: If f(a) = b, then f^{-1}(b) = a.
  • Composition: f(f^{-1}(x)) = f^{-1}(f(x)) = x for all x in the domain of either function.
  • Graphically: f^{-1}(x) is the reflection of f(x) over the line y = x.
  • Algebraically: To find f^{-1}(x), switch x and y in the equation for f(x) and solve for y.
  • Verbally: f^{-1}(x) uses the opposite operations of f(x) in reverse order.
  • A function f(x) is invertible if every output value of f(x) comes from a unique input value.
  • A function that is strictly increasing or strictly decreasing is invertible.
  • The domain of f^{-1}(x) is the range of f(x), and vice-versa.

Circular Trigonometry

  • \theta: Standard position angle
  • R: Reference angle (positive acute angle from terminal side to x-axis).
  • Trigonometric Functions:
    • \sin \theta = \frac{y}{r}
    • \cos \theta = \frac{x}{r}
    • \tan \theta = \frac{y}{x}
  • Unit Circle: r = 1
    • \sin \theta: y-coordinate of point P
    • \cos \theta: x-coordinate of point P
    • \tan \theta: Slope of terminal side

Sine/Cosine Graph Properties

  • General form: f(x) = a \sin(b(x+c)) + d
  • y = \sin x: Starts at midline and goes upward.
  • y = \cos x: Starts at maximum and goes downward.
  • |a|: Amplitude
  • d: Midline
  • -c: Phase shift
  • \frac{2\pi}{b}: Period
  • |a| + d: Maximum
  • |a| - d: Minimum

Identities

  • Pythagorean Identities:
    • \sin^2 \theta + \cos^2 \theta = 1
    • \tan^2 \theta + 1 = \sec^2 \theta
    • \cot^2 \theta + 1 = \csc^2 \theta
  • Reciprocal Identities:
    • \csc \theta = \frac{1}{\sin \theta}
    • \sec \theta = \frac{1}{\cos \theta}
    • \cot \theta = \frac{1}{\tan \theta}
  • Quotient Identities:
    • \tan \theta = \frac{\sin \theta}{\cos \theta}
    • \cot \theta = \frac{\cos \theta}{\sin \theta}
  • Angle Sum/Difference Identities:
    • \sin(a+b) = \sin a \cos b + \cos a \sin b
    • \sin(a-b) = \sin a \cos b - \cos a \sin b
    • \cos(a+b) = \cos a \cos b - \sin a \sin b
    • \cos(a-b) = \cos a \cos b + \sin a \sin b
  • Double Angle Identities:
    • \sin 2\theta = 2 \sin \theta \cos \theta
    • \cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2 \cos^2 \theta - 1 = 1 - 2 \sin^2 \theta

Solving Trig Equations

  • \sin \theta = m (y-coordinate)
  • \cos \theta = n (x-coordinate)
  • \tan \theta = p (slope)
  • Start by isolating the trig function.
  • To find all solutions, take each answer \pm 2\pi k for any integer value of K.

Polar Coordinates

  • (r, \theta)
  • Rectangular to Polar Conversion:
    • r^2 = x^2 + y^2
    • \theta = \arctan(\frac{y}{x})
  • Polar to Rectangular Conversion:
    • x = r \cos \theta
    • y = r \sin \theta
  • Polar form of a complex number:
    • a + bi = r(\cos \theta + i \sin \theta)
  • If r is negative, it is plotted in the opposite direction of \theta.

Inverse Trig Functions

  • f(x) = \sin^{-1} x or \arcsin(x)
    • Input: Ratio of sides or y-coordinate on the unit circle.
    • Output: Standard position angle.
    • Domain: x \in [-1, 1]
    • Range: y \in [-\frac{\pi}{2}, \frac{\pi}{2}] (Quadrants 1 & 4)
  • g(x) = \cos^{-1} x or \arccos(x)
    • Input: Ratio of sides or x-coordinate on the unit circle.
    • Output: Standard position angle.
    • Domain: x \in [-1, 1]
    • Range: y \in [0, \pi] (Quadrants 1 & 2)
  • h(x) = \tan^{-1} x or \arctan(x)
    • Input: Ratio of sides or slope of the terminal side of \theta.
    • Output: Standard position angle.
    • Domain: x \in (-\infty, \infty)
    • Range: y \in (-\frac{\pi}{2}, \frac{\pi}{2}) (Quadrants 1 & 4)
  • \tan^{-1} (\frac{y}{x}) = Ref. angle.
  • Use Quadrant and Ref angle to find \theta