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 $\theta$