# Trigonometric Identities and Equations

## Modeling with Trigonometric Functions

Suppose we charted the average daily temperatures in New York City over the course of one year. We would expect to find the lowest temperatures in January and February and highest in July and August. This familiar cycle repeats year after year, and if we were to extend the graph over multiple years, it would resemble a periodic function.
Many other natural phenomena are also periodic. For example, the phases of the moon have a period of approximately 28 days, and birds know to fly south at about the same time each year.
So how can we model an equation to reflect periodic behavior? First, we must collect and record data. We then find a function that resembles an observed pattern. Finally, we make the necessary alterations to the function to get a model that is dependable. In this section, we will take a deeper look at specific types of periodic behavior and model equations to fit data.

### Determining the Amplitude and Period of a Sinusoidal Function

Any motion that repeats itself in a fixed time period is considered periodic motion and can be modeled by a sinusoidal function. The amplitude of a sinusoidal function is the distance from the midline to the maximum value, or from the midline to the minimum value. The midline is the average value. Sinusoidal functions oscillate above and below the midline, are periodic, and repeat values in set cycles. Recall from Graphs of the Sine and Cosine Functions that the period of the sine function and the cosine function is 
  In other words, for any value of 
  

### Finding Equations and Graphing Sinusoidal Functions

One method of graphing sinusoidal functions is to find five key points. These points will correspond to intervals of equal length representing 
  of the period. The key points will indicate the location of maximum and minimum values. If there is no vertical shift, they will also indicate x-intercepts. For example, suppose we want to graph the function 
  We know that the period is 
  so we find the interval between key points as follows.
Starting with 
  we calculate the first y-value, add the length of the interval 
  to 0, and calculate the second y-value. We then add 
  repeatedly until the five key points are determined. The last value should equal the first value, as the calculations cover one full period. Making a table similar to , we can see these key points clearly on the graph shown in .

### Modeling Periodic Behavior

We will now apply these ideas to problems involving periodic behavior.

### Modeling Harmonic Motion Functions

Harmonic motion is a form of periodic motion, but there are factors to consider that differentiate the two types. While general periodic motion applications cycle through their periods with no outside interference, harmonic motion requires a restoring force. Examples of harmonic motion include springs, gravitational force, and magnetic force.

### Simple Harmonic Motion

A type of motion described as simple harmonic motion involves a restoring force but assumes that the motion will continue forever. Imagine a weighted object hanging on a spring, When that object is not disturbed, we say that the object is at rest, or in equilibrium. If the object is pulled down and then released, the force of the spring pulls the object back toward equilibrium and harmonic motion begins. The restoring force is directly proportional to the displacement of the object from its equilibrium point. When 
  
 

### Damped Harmonic Motion

In reality, a pendulum does not swing back and forth forever, nor does an object on a spring bounce up and down forever. Eventually, the pendulum stops swinging and the object stops bouncing and both return to equilibrium. Periodic motion in which an energy-dissipating force, or damping factor, acts is known as damped harmonic motion. Friction is typically the damping factor.
In physics, various formulas are used to account for the damping factor on the moving object. Some of these are calculus-based formulas that involve derivatives. For our purposes, we will use formulas for basic damped harmonic motion models.

### Bounding Curves in Harmonic Motion

Harmonic motion graphs may be enclosed by bounding curves. When a function has a varying amplitude, such that the amplitude rises and falls multiple times within a period, we can determine the bounding curves from part of the function.

### Key Equations


### Key Concepts

1. Sinusoidal functions are represented by the sine and cosine graphs. In standard form, we can find the amplitude, period, and horizontal and vertical shifts. See  and .
2. Use key points to graph a sinusoidal function. The five key points include the minimum and maximum values and the midline values. See .
3. Periodic functions can model events that reoccur in set cycles, like the phases of the moon, the hands on a clock, and the seasons in a year. See , ,  and .
4. Harmonic motion functions are modeled from given data. Similar to periodic motion applications, harmonic motion requires a restoring force. Examples include gravitational force and spring motion activated by weight. See .
5. Damped harmonic motion is a form of periodic behavior affected by a damping factor. Energy dissipating factors, like friction, cause the displacement of the object to shrink. See , , , , and .
6. Bounding curves delineate the graph of harmonic motion with variable maximum and minimum values. See .

### Section Exercises


### Verbal


### Algebraic 

For the following exercises, find a possible formula for the trigonometric function represented by the given table of values.

### Graphical

For the following exercises, graph the given function, and then find a possible physical process that the equation could model.

### Technology

For the following exercise, construct a function modeling behavior and use a calculator to find desired results.

### Real-World Applications

For the following exercises, construct a sinusoidal function with the provided information, and then solve the equation for the requested values.
For the following exercises, find the amplitude, period, and frequency of the given function.
For the following exercises, construct an equation that models the described behavior.
For the following exercises, construct an equation that models the described behavior.
For the following exercises, create a function modeling the described behavior. Then, calculate the desired result using a calculator.

### Extensions

For the following exercises, find a function of the form 
  that fits the given data.
For the following exercises, find a function of the form 
   that fits the given data.

### Chapter Review Exercises


### Solving Trigonometric Equations with Identities

For the following exercises, find all solutions exactly that exist on the interval 
    
   
For the following exercises, use basic identities to simplify the expression.
For the following exercises, determine if the given identities are equivalent.

### Sum and Difference Identities

For the following exercises, find the exact value.
For the following exercises, prove the identity.
For the following exercise, simplify the expression.
For the following exercises, find the exact value.

### Double-Angle, Half-Angle, and Reduction Formulas

For the following exercises, find the exact value.
For the following exercises, use  to find the desired quantities.
For the following exercises, prove the identity.
For the following exercises, rewrite the expression with no powers.

### Sum-to-Product and Product-to-Sum Formulas

For the following exercises, evaluate the product for the given expression using a sum or difference of two functions. Write the exact answer.
For the following exercises, evaluate the sum by using a product formula. Write the exact answer.
For the following exercises, change the functions from a product to a sum or a sum to a product.

### Solving Trigonometric Equations

For the following exercises, find all exact solutions on the interval 
    
   
For the following exercises, find all exact solutions on the interval 
    
   
For the following exercises, simplify the equation algebraically as much as possible. Then use a calculator to find the solutions on the interval 
     Round to four decimal places.
For the following exercises, graph each side of the equation to find the zeroes on the interval 
    
   

### Modeling with Trigonometric Equations

For the following exercises, graph the points and find a possible formula for the trigonometric values in the given table.
For the following exercises, construct functions that model the described behavior.
For the following exercises, find the amplitude, frequency, and period of the given equations.
For the following exercises, model the described behavior and find requested values.

### Practice Test

For the following exercises, simplify the given expression.
For the following exercises, find the exact value.
For the following exercises, find all exact solutions to the equation on 
   
  
For the following exercises, prove the identity.