# Further Applications of Trigonometry

## Parametric Equations: Graphs

While not every fan (or team manager) appreciates it, baseball and many other sports have become dependent on analytics, which involve complex data recording and quantitative evaluation used to understand and predict behavior. The earliest influence of analytics was mostly statistical; more recently, physics and other sciences have come into play. Foremost among these is the focus on launch angle and exit velocity, which when at certain values can almost guarantee a home run. On the other hand, emphasis on launch angle and focusing on home runs rather than overall hitting results in far more outs. Consider the following situation: it is the bottom of the ninth inning, with two outs and two players on base. The home team is losing by two runs. The batter swings and hits the baseball at 140 feet per second and at an angle of approximately 
  to the horizontal. How far will the ball travel? Will it clear the fence for a game-winning home run? The outcome may depend partly on other factors (for example, the wind), but mathematicians can model the path of a projectile and predict approximately how far it will travel using parametric equations. In this section, we’ll discuss parametric equations and some common applications, such as projectile motion problems.

### Graphing Parametric Equations by Plotting Points

In lieu of a graphing calculator or a computer graphing program, plotting points to represent the graph of an equation is the standard method. As long as we are careful in calculating the values, point-plotting is highly dependable.

### Applications of Parametric Equations

Many of the advantages of parametric equations become obvious when applied to solving real-world problems. Although rectangular equations in x and y give an overall picture of an object's path, they do not reveal the position of an object at a specific time. Parametric equations, however, illustrate how the values of x and y change depending on t, as the location of a moving object at a particular time.
A common application of parametric equations is solving problems involving projectile motion. In this type of motion, an object is propelled forward in an upward direction forming an angle of 
            to the horizontal, with an initial speed of 
           and at a height 
            above the horizontal.
The path of an object propelled at an inclination of 
              to the horizontal, with initial speed 
  and at a height 
              above the horizontal, is given by
where 
             accounts for the effects of gravity and 
              is the initial height of the object. Depending on the units involved in the problem, use 
  or 
  The equation for 
             gives horizontal distance, and the equation for 
              gives the vertical distance.

### Key Concepts

1. When there is a third variable, a third parameter on which 
           and 
           depend, parametric equations can be used.
2. To graph parametric equations by plotting points, make a table with three columns labeled 
           and 
           Choose values for 
           in increasing order. Plot the last two columns for 
           and 
           See  and .
3. When graphing a parametric curve by plotting points, note the associated t-values and show arrows on the graph indicating the orientation of the curve. See  and .
4. Parametric equations allow the direction or the orientation of the curve to be shown on the graph. Equations that are not functions can be graphed and used in many applications involving motion. See .
5. Projectile motion depends on two parametric equations: 
  and 
  Initial velocity is symbolized as 
   represents the initial angle of the object when thrown, and 
            represents the height at which the object is propelled.

### Section Exercises


### Verbal


### Graphical

For the following exercises, graph each set of parametric equations by making a table of values. Include the orientation on the graph.
For the following exercises, sketch the curve and include the orientation.
For the following exercises, graph the equation and include the orientation. Then, write the Cartesian equation.
For the following exercises, graph the equation and include the orientation.
For the following exercises, use the parametric equations for integers a and b:
For the following exercises, describe the graph of the set of parametric equations.
For the following exercises, use a graphing utility to graph on the window 
     by 
     on the domain 
     for the following values of 
     and 
    , and include the orientation.
  


### Technology

For the following exercises, look at the graphs that were created by parametric equations of the form 
  Use the parametric mode on the graphing calculator to find the values of 
              and 
              to achieve each graph.
For the following exercises, use a graphing utility to graph the given parametric equations.
        
          

### Extensions

For the following exercises, use this scenario: A dart is thrown upward with an initial velocity of 64 ft/s at an angle of elevation of 52°. Consider the position of the dart at any time 
             Neglect air resistance.
For the following exercises, look at the graphs of each of the four parametric equations. Although they look unusual and beautiful, they are so common that they have names, as indicated in each exercise. Use a graphing utility to graph each on the indicated domain.