Study Guide: Graphs and Equations of Trigonometric Functions (Period, Amplitude, Midline, Transformations, and Solving Trig Equations)

Graphs and Transformations of Sine and Cosine Functions

Period, Amplitude, Midline, and Transformations

Trigonometric functions such as sine and cosine can be transformed by changing their amplitude, period, phase shift (horizontal shift), and midline (vertical shift). Understanding these transformations is essential for graphing and analyzing trigonometric functions.

• Amplitude: The amplitude of a sine or cosine function is the distance from the midline to the maximum (or minimum) value of the function. For $y = A ext{sin}(Bx + C) + D$ or $y = A ext{cos}(Bx + C) + D$, the amplitude is $|A|$.

• Period: The period is the length of one complete cycle of the function. For $y = A ext{sin}(Bx + C) + D$ or $y = A ext{cos}(Bx + C) + D$, the period is $\frac{2\pi}{|B|}$.

• Midline: The midline is the horizontal line that runs through the center of the graph. For $y = A ext{sin}(Bx + C) + D$, the midline is $y = D$.

• Horizontal Shift (Phase Shift): The horizontal shift is given by $-\frac{C}{B}$.

Example: For $y = 2\sin(4(x - \frac{\pi}{2})) + 1$:

• Amplitude: $2$

• Period: $\frac{2\pi}{4} = \frac{\pi}{2}$

• Horizontal Shift: $\frac{\pi}{2}$ to the right

• Midline: $y = 1$

Example: For $y = 3\sin(x) - \frac{2}{3}$:

• Amplitude: $3$

• Period: $2\pi$

• Horizontal Shift: $0$

• Midline: $y = -\frac{2}{3}$

Graphing Trigonometric Functions

To graph a transformed sine or cosine function:

1. Identify amplitude, period, midline, and phase shift.

2. Mark the midline on the graph.

3. Plot key points: maximum, minimum, and intercepts based on the transformations.

4. Sketch one or more cycles using the identified period.

Application: Trigonometric functions are used to model periodic phenomena, such as population cycles, sound waves, and seasonal changes.

Solving Trigonometric Equations

General Approach

Solving trigonometric equations involves finding all angles that satisfy a given equation within a specified interval. The process often uses properties of periodicity, reference angles, and the unit circle.

• Isolate the trigonometric function on one side of the equation.

• Find the reference angle for the given value.

• Determine all solutions within the specified interval, considering the sign and quadrant.

• Use the unit circle to identify all possible solutions.

Example: Solve $\sin(\theta) = \frac{\sqrt{2}}{2}$ for $\theta$ in $[0, 2\pi]$.

• Reference angle: $\frac{\pi}{4}$

• Solutions: $\theta = \frac{\pi}{4}, \frac{3\pi}{4}$

Example: Solve $2\sin(2\theta) - 1 = 0$ for $\theta$ in $[0, 2\pi]$.

• Isolate: $\sin(2\theta) = \frac{1}{2}$

• Reference angle: $\frac{\pi}{6}$

• General solutions for $2\theta$: $\frac{\pi}{6}, \frac{5\pi}{6}$

• Divide by 2: $\theta = \frac{\pi}{12}, \frac{5\pi}{12}$

• Also consider $2\theta$ values in $[0, 4\pi]$ for all solutions in $[0, 2\pi]$.

Reference Angles and Quadrants

The sign of sine and cosine depends on the quadrant:

• Quadrant I: $\sin(\theta) > 0$, $\cos(\theta) > 0$

• Quadrant II: $\sin(\theta) > 0$, $\cos(\theta) < 0$

• Quadrant III: $\sin(\theta) < 0$, $\cos(\theta) < 0$

• Quadrant IV: $\sin(\theta) < 0$, $\cos(\theta) > 0$

Reference angles are used to find solutions in all relevant quadrants.

Solving Quadratic Trigonometric Equations

Some trigonometric equations are quadratic in form and can be solved by factoring:

• Example: $2\sin^2(\theta) - 3\sin(\theta) + 1 = 0$

• Factor: $(2\sin(\theta) - 1)(\sin(\theta) - 1) = 0$

• Solutions: $\sin(\theta) = \frac{1}{2}$ or $\sin(\theta) = 1$

• Find all $\theta$ in $[0, 2\pi]$ that satisfy these equations.

Table: Integer Values for k in General Solution

When solving equations like $\sin(2\theta) = \frac{1}{2}$, the general solution is $2\theta = \frac{\pi}{6} + 2k\pi$ and $2\theta = \frac{5\pi}{6} + 2k\pi$ for integer $k$. Plugging in integer values for $k$ gives all solutions in the interval.

k	$\theta = \frac{\pi}{12} + k\pi$	$\theta = \frac{5\pi}{12} + k\pi$
0	$\frac{\pi}{12}$	$\frac{5\pi}{12}$
1	$\frac{13\pi}{12}$	$\frac{17\pi}{12}$
2	$\frac{25\pi}{12}$	$\frac{29\pi}{12}$

Additional info: Values outside the interval $[0, 2\pi]$ are excluded from the solution set.

Modeling Periodic Phenomena with Trigonometric Functions

Applications: Population Cycles

Trigonometric functions are used to model periodic changes in real-world phenomena, such as animal populations that fluctuate in cycles. The graphs of fox and rabbit populations over time can be approximated by sine or cosine functions, reflecting their regular, repeating patterns.

• Appropriateness: The periodic nature of the population data matches the periodicity of sine and cosine functions.

• Modeling: Equations of the form $y = A\sin(Bx + C) + D$ or $y = A\cos(Bx + C) + D$ can be fitted to the data, where $A$ is amplitude, $B$ relates to period, $C$ is phase shift, and $D$ is midline.

Example: If the rabbit population oscillates between 300 and 1300 over a 24-month period, the amplitude is 500, the midline is 800, and the period is 24 months.

Summary: Key Concepts

• The sign of sine and cosine is determined by the quadrant in which the angle lies.

• Reference angles help find all solutions to trigonometric equations by relating the given value to a standard angle in the first quadrant.

• Solving trigonometric equations differs from solving algebraic equations due to periodicity and the need to consider all possible solutions within the interval.