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  2. AP Pre Calculus
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Formula relating cosine and sine.

cos(θ)=sin(θ+π2)cos(θ) = sin(θ + \frac{π}{2})cos(θ)=sin(θ+2π​)

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Formula relating cosine and sine.

cos(θ)=sin(θ+π2)cos(θ) = sin(θ + \frac{π}{2})cos(θ)=sin(θ+2π​)

Formula for frequency (f).

f=1Tf = \frac{1}{T}f=T1​

Formula for midline (k).

k=ymax+ymin2k = \frac{y_{max} + y_{min}}{2}k=2ymax​+ymin​​

Formula for amplitude.

Amplitude=ymax−ymin2Amplitude = \frac{y_{max} - y_{min}}{2}Amplitude=2ymax​−ymin​​

Formula for odd symmetry.

f(−x)=−f(x)f(-x) = -f(x)f(−x)=−f(x)

Formula for even symmetry.

f(−x)=f(x)f(-x) = f(x)f(−x)=f(x)

Period of standard sine function.

2π2\pi2π

Period of standard cosine function.

2π2\pi2π

Frequency of standard sine function.

12π\frac{1}{2\pi}2π1​

Frequency of standard cosine function.

12π\frac{1}{2\pi}2π1​

Define sinusoidal function.

A function resembling sine or cosine, oscillating and periodic.

What is the period (T) of a sinusoidal function?

The length of one complete cycle.

Define frequency (f).

Number of cycles per unit time; reciprocal of period.

What is the midline (k)?

The horizontal line that cuts the wave in half.

Define amplitude.

Vertical distance from midline to max/min value.

What is odd symmetry?

Symmetry about the origin; f(-x) = -f(x).

What is even symmetry?

Symmetry about the y-axis; f(-x) = f(x).

Define oscillation in sinusoidal functions.

Up and down movement between two values, creating a wave.

What does concave up mean?

The curve looks like a smile.

What does concave down mean?

The curve looks like a frown.

Sine vs. Cosine: Symmetry.

Sine: Odd | Cosine: Even

Sine vs. Cosine: Shift.

Cosine is sine shifted left by π2\frac{π}{2}2π​.

Amplitude vs. Midline.

Amplitude: Vertical distance from midline | Midline: Horizontal center.

Period vs. Frequency.

Period: Length of one cycle | Frequency: Cycles per unit time.

Concave Up vs. Concave Down.

Concave Up: Curve smiles | Concave Down: Curve frowns.

What are the differences between period and frequency?

Period: Time for one cycle | Frequency: Cycles per unit time

What are the differences between amplitude and midline?

Amplitude: Vertical distance from midline | Midline: Horizontal center of the wave

What are the differences between sine and cosine functions?

Sine: Starts at midline | Cosine: Starts at maximum/minimum

What are the differences between a vertical stretch and a horizontal stretch?

Vertical: Affects amplitude | Horizontal: Affects period

What are the differences between positive and negative amplitudes?

Positive: Starts increasing | Negative: Starts decreasing