Study Notes: Period, Amplitude, Midline, and Solving Trigonometric Equations

Graphs and Transformations of Sine and Cosine Functions

Understanding Period, Amplitude, and Midline

Trigonometric functions such as sine and cosine can be transformed by changing their amplitude, period, phase shift, and midline. These transformations allow us to model a wide variety of periodic phenomena.

• 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 a function of the form $y = A \cos(Bx + C) + D$, the amplitude is $|A|$.

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

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

• Phase Shift: The horizontal shift of the graph, given by $-\frac{C}{B}$.

Example: For $y = 2\cos(3x - \frac{\pi}{2}) + 1$:

• Amplitude: $2$

• Period: $\frac{2\pi}{3}$

• Phase Shift: $\frac{\pi}{6}$ to the right

• Midline: $y = 1$

Graphing Sine and Cosine with Transformations

To graph a transformed sine or cosine function:

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

2. Mark the midline on the graph.

3. Plot key points: start, maximum, midline crossing, minimum, and end of one period.

4. Repeat the pattern for additional cycles as needed.

Example: $y = 3\sin(x - \frac{\pi}{2}) - 1$

• Amplitude: $3$

• Period: $2\pi$

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

• Midline: $y = -1$

Modeling with Trigonometric Functions

Applications: Population Cycles

Periodic phenomena, such as animal populations, can be modeled using sine and cosine functions. For example, the population of foxes and rabbits in a park over time may oscillate in a predictable, periodic manner.

• Choose a trigonometric function (sine or cosine) that matches the observed pattern.

• Adjust amplitude, period, phase shift, and midline to fit the data.

Example: If the rabbit population oscillates between 800 and 1200 with a period of 12 months, a possible model is:

• Midline: $y = 1000$

• Amplitude: $200$

• Period: $12$(so$B = \frac{2\pi}{12} = \frac{\pi}{6}$)

• Equation: $y = 200\sin\left(\frac{\pi}{6}t\right) + 1000$

Solving Trigonometric Equations

Review of Sine and Cosine Properties

Before solving trigonometric equations, recall the following:

• Signs in Quadrants: The sign of $\sin(\theta)$ and $\cos(\theta)$ depends on the quadrant in which the angle lies.

• Reference Angles: The reference angle is the acute angle formed with the x-axis. It is used to find the values of trigonometric functions for any angle.

Table: Signs of Sine and Cosine in Each Quadrant

Quadrant	sin(θ)	cos(θ)
I	+	+
II	+	-
III	-	-
IV	-	+

Solving Basic Trigonometric Equations

To solve equations such as $\sin(\theta) = a$ or $\cos(\theta) = b$:

1. Isolate the trigonometric function.

2. Find the reference angle.

3. Determine all solutions in the given interval, considering the sign and period of the function.

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

• Reference angle: $\frac{\pi}{4}$

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

Solving Quadratic Trigonometric Equations

Some equations may be quadratic in form, such as $2\sin^2(\theta) - 3\sin(\theta) + 1 = 0$.

1. Let $u = \sin(\theta)$ and solve the quadratic equation for $u$.

2. Solve for $\theta$ for each solution of $u$.

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

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

• Factor: $(2u - 1)(u - 1) = 0$

• So $u = \frac{1}{2}$ or $u = 1$

• Thus, $\sin(\theta) = \frac{1}{2}$ or $\sin(\theta) = 1$

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

Using the Unit Circle

The unit circle is a powerful tool for solving trigonometric equations. It helps identify all angles that correspond to a given sine or cosine value.

• For $\sin(\theta) = a$, find all angles where the y-coordinate is $a$.

• For $\cos(\theta) = b$, find all angles where the x-coordinate is $b$.

Summary Table: Integer Values for $k$ in General Solutions

k	Angle 1	Angle 2
0	$\frac{\pi}{4}$	$\frac{3\pi}{4}$
1	$\frac{9\pi}{4}$	$\frac{11\pi}{4}$
2	$\frac{17\pi}{4}$	$\frac{19\pi}{4}$
...	...	...

Additional info: Table entries are inferred from the context of solving $\sin(\theta) = \frac{\sqrt{2}}{2}$ and generalizing solutions using $2\pi k$.

Practice Problems

• Find the amplitude, period, horizontal shift, and midline for $y = 2\cos(4(x - \frac{\pi}{2})) + 1$.

• Find all solutions to $\sin(2\theta) = 1$ for $0 \leq \theta < 2\pi$.

• Solve $2\sin^2(\theta) - 3\sin(\theta) + 1 = 0$ for $0 \leq \theta < 2\pi$.

Summary

• The sign of $\sin(\theta)$ and $\cos(\theta)$ depends on the quadrant.

• Reference angles help relate any angle to an acute angle in the first quadrant.

• Solving trigonometric equations often involves finding all solutions in a given interval, using properties of periodicity and symmetry.