Pythagorean Identities

Pythagorean Identity 1: sin^2(x)+cos^2(x)=1

Pythagorean Identity 2: tan^2(x)+1=sec^2(x)

Pythagorean Identity 3: 1+cot^2(x)=csc^2(x)

Equation

y=Asin(B(x±C))±D

A: Amplitude

B: Frequency

C: Phase shift/vertical shift

D: Midline/horizontal shift

Vocabulary

Wavelength/period: The distance it takes to complete one cycle/wave.

Frequency: The number of full periods that fit between 0 and 2π

Asymptote: A line that the function approaches but never reaches. This happens when the function is undefined.

Graphs

y=sin(x): Wavy shape, curve passes through at midline at y-axis

y=cos(x): Wavy shape, curve passes through at maximum at y-axis

y=tan(x): “Chairs” facing negative numbers, one “chair” passes through midline at y-axis

y=csc(x): Us and upside down Us, y-axis is asymptote

y=sec(x): Us and upside down Us, y-axis is NOT asymptote

y=cot(x): “Chairs” facing positive numbers, no “chair” passes through midline at y-axis

y=sin^-1(x): Axes switched, single “chair” facing negative numbers, “chair” passes through midline at y-axis

y=cos^-1(x): Axes switched, single “chair” facing positive numbers, “chair” does NOT passes through midline at y-axis

y=tan^-1(x): Axes switched, “tipped over chair”, “chair” passes through midline at y-axis