Trigonometric and Polar Functions

The period increases by a factor of three from $2\\pi$ to $6\\pi$.

The period decreases by a factor of three from $\\pi$ to $\\frac{\\pi}{3}$.

There is no change in period; it remains at $2\\pi$.

The period decreases by a factor of three from $2\\pi$ to $\\frac{2\\pi}{3}$.

$\geq 7-4\sqrt{12}$

$<7-4\sqrt{12}$

$\leq 5-\sqrt[3]{125}$

$>9-8\sqrt[4]{27}$

$y=\sec\left(\frac{1}{2}x+\frac{\pi}{4}\right)$

$y=\sec\left(\frac{x}{2}-\frac{\pi}{8}\right)$

$y=0.5(\sec)(x-\frac{\pi}{4})$

$y=\sec(4x-\frac{\pi}{2})$

$\text{x} = \frac{\pi}{2}$

$\text{x} = 0$

$\text{x} = -\frac{\pi}{2}$

$\text{x} = \pi$

Inverse square relationship shown by a transformation of a secant function.

Periodicity increase and decreases corresponding to sine wave amplitudes for varying levels of sound.

Constant values along a cosecant wave indicating static audio volume regardless of distance.

Zero points of a cosecant function representing moments of silence between sound signals.

2

-0.5

0.5

-2

sin(x), since sine is reciprocal to cosecant which could be undefined when cotangent is zero.

sec(x), as secant involves cosine which relates directly with cotangent values being zeros through reciprocal identities.

None; no other trigonometric ratios are zero when $\cot(x)=0$.

tan(x), because tangent and cotangent can both be simultaneously zero.

√2

1

0

Undefined

$\frac{17}{10}$

$\frac{24}{7}$

$\frac{10}{7}$

$\frac{17}{24}$

45º ,135º ,225º ,315º

0º ,90º ,180º ,270º

60º ,120º ,240º

30º ,150º ,210º ,330º