Sinusoidal functions model periodic real-world phenomena (tides, temperature, population cycles, sound waves). They follow one of two standard forms:
f(x) = A\sin\left(B(x - C)\right) + D \\ f(x) = A\cos\left(B(x - C)\right) + D
$|A|$ = amplitude: vertical distance from midline to peak/trough
$B$ = angular frequency: period (length of one full cycle) = $\frac{2\pi}{|B|}$
$C$ = phase shift: horizontal shift (right if $C>0$, left if $C<0$)
$D$ = vertical shift: midline of the function at $y=D$
3. Inverse Trigonometric Functions★★★☆☆⏱ 3 min
Trigonometric functions are not one-to-one over their full domain, so we restrict their domains to define valid inverse functions that return a single unique angle:
Key property: $
\\sin(\\arcsin(x)) = x$ for all $x \in [-1,1]$, but $
\\arcsin(\\sin(x)) = x$ *only* if $x$ falls within the restricted range of $
\\arcsin(x)$. Adjust for angles outside this range.
4. Polar Coordinates and Common Curves★★★☆☆⏱ 4 min
Polar coordinates represent points as $(r, \theta)$ instead of Cartesian $(x,y)$, where $r$ is the signed distance from the origin (pole), and $ heta$ is the angle from the positive x-axis (polar axis). A negative $r$ means you move $|r|$ units in the opposite direction of $ heta$.
x = r\cos\theta, \quad y = r\sin\theta \\ r^2 = x^2 + y^2, \quad \tan\theta = \frac{y}{x}
**Circles**: $r = a$ (radius $a$ at origin), $r = 2a\\cos\theta$ (radius $a$ centered at $(a, 0)$), $r = 2a\\sin\theta$ (radius $a$ centered at $(0,a)$)
**Cardioids**: $r = a(1 \pm \\cos\theta)$ or $r = a(1 \pm \\sin\theta)$: heart-shaped, cusp at the origin
**Rose curves**: $r = a\\cos(n\theta)$ or $r = a\\sin(n\theta)$: $n$ petals if $n$ is odd, $2n$ petals if $n$ is even, petal length $|a|$
Common Pitfalls
Why: Most students only memorize reference angle magnitude values and skip the quadrant check after calculating the value
Why: Students confuse horizontal shift rules with vertical shift rules, and often forget to factor out $B$ from the argument
Why: Students find any angle that matches the trigonometric ratio, rather than the unique angle required for the inverse function
Why: Students apply the same petal count rule for both odd and even values of $n$
Why: Most students assume $r$ is always a positive distance from the pole