Trigonometric and Polar Functions

The left and right limits exist and are finite but unequal at that point.

There are no limits because the function becomes non-real at that point.

The left and right limits agree yet differ from the function’s value at that point.

Both one-sided limits tend towards infinity but in opposite directions.

4π

2π

π

π/2

-5

5

-3

3

Degrees

Cubic centimeters

Meters

Radians

Sine wave (e.g., $h(t)=5\sin(\theta)$)

Polynomial (e.g., $f(x)=x^2$)

Tangent (e.g., $g(t)=\tan(0)$)

Quadratic (e.g., $f(t)=\sin t+5$)

$\tan(\tan^{-1}(t))$

$\cos(\cos^{-1}(t))$

$\sec(\sec^{-1}(t))$

$\sin(\sin^{-1}(t))$

It shifts the graph horizontally right/left depending on the sign of $c$.

It flattens or compresses the vertical stretch, thus making the waves appear less tall.

It heightens the entire wave vertically, increasing the distance between the midline and maximum/minimum values.

It zooms in or out and changes both the amplitude and period simultaneously.

$\\sec^2(x)$

$\\cos^2(x)$

$\\tan^2(x)$

$\\cot^2(x)$

Inverts graph

Shifts graph left or right

Shrinks graph horizontally

Stretches graph vertically

Because $h^{-1}(x)=\sin^{-1}(\ln(x))$ isn't periodic like $h(x)$.

Because $\ln(x)$ only exists for $x > 0$.

Because $e^{\sin(x)}$ isn't one-to-one without proper domain restriction.

Because $\sin^{-1}(\ln(x))$ only exists for $-\pi \leq x \leq \pi$.