3.7 Trigonometric Functions

Graphing trigonometric functions creates repeating waves.

The Sine Function ($f(x) = \sin x$):

The domain is all real numbers ($x \in \mathbb{R}$). The range is $y \in [-1, 1]$.
The center line is the x-axis ($y = 0$).
The amplitude is $1$.
The period is $2\pi$.

The Cosine Function ($f(x) = \cos x$):

Has the same domain, range, center line, amplitude, and period as the sine function.
The cosine wave is the sine wave shifted horizontally by $\dfrac{\pi}{2}$.

The Tangent Function ($f(x) = \tan x$):

The range is all real numbers ($y \in \mathbb{R}$). Amplitude does not apply.
The domain excludes values where the function is undefined, creating vertical asymptotes at $x = \pm\dfrac{\pi}{2}, \pm\dfrac{3\pi}{2}$, etc.
The period is $\pi$.

Trigonometric waves are modified through mathematical adjustments to the base function.

The format $f(x) = A\sin(B(x-D)) + C$ (and similarly for cosine) defines these graphical changes:

Vertical Shift ($C$): The center line is $y = C$.
Amplitude ($|A|$): The distance from the center line to the maximum or minimum. The maximum is $C + |A|$ and the minimum is $C - |A|$.
Period Modifier ($B$): The period is $T = \dfrac{2\pi}{B}$, or $B = \dfrac{2\pi}{T}$.
Horizontal Shift ($D$): Shifts the graph to the right by $D$ units.

Note: For the tangent function $f(x) = A\tan(B(x-D)) + C$, the period is $T = \dfrac{\pi}{B}$. Amplitude does not apply, but $A$ affects steepness.

Analyzing graphical properties helps determine the mathematical equation.

A sine wave has a minimum of $10$, a maximum of $20$, and a period of $8$.

The center line is: $C = \dfrac{20 + 10}{2} \implies \mathbf{C = 15}$.

The amplitude is: $A = 20 - 15 \implies \mathbf{A = 5}$.

The period determines $B$: $B = \dfrac{2\pi}{8} \implies \mathbf{B = \dfrac{\pi}{4}}$.

The equation is: $\mathbf{f(x) = 5\sin\left(\dfrac{\pi}{4}x\right) + 15}$.

A cosine wave has a minimum of $-10$, a maximum of $20$, and a period of $\pi$.

The center line is: $C = \dfrac{20 + (-10)}{2} \implies \mathbf{C = 5}$.

The amplitude is: $A = 20 - 5 \implies \mathbf{A = 15}$.

The period determines $B$: $B = \dfrac{2\pi}{\pi} \implies \mathbf{B = 2}$.

The equation is: $\mathbf{f(x) = 15\cos(2x) + 5}$.

A wave can be represented by multiple equations by changing the horizontal shift ($D$) to start at different points:

Graph $f(x) = 5\sin(2x) + 7$ for $0 \le x \le 2\pi$.

The center line is $y = 7$, the maximum is $12$, and the minimum is $2$.

The period is $T = \dfrac{2\pi}{2} = \pi$. Graphing up to $2\pi$ requires two complete cycles.

Start on the center line and move upwards to graph the positive sine function.