CK-12 Trigonometry Read online

  a. To evaluate the function, you can “trace” on the graph. Press the [TRACE] button. You should see the equation at the top of the screen, and the cursor should be on the y-intercept, . At the bottom of the screen you should see and .This tells us that for , the function value is .

  Now that you are in tracing mode, you can enter any value, and the calculator will tell you the value. For example, if you press [2 ENTER], you will see the cursor move to the point and at the bottom of the screen, you will see and . If you press -[2 ENTER], you will see and at the bottom of the screen. Notice that you cannot see the point on the graph. To see that point, we need to change the window. Press [WINDOW] and scroll down to Ymin. Change the to . Then press [GRAPH]. Now press [TRACE] -[2 ENTER]. You should see the point .

  b. End behavior: the left-hand side of the graph appears to be going down, and the right-hand side appears to be going up. If we want to see more of the graph, we can zoom out. Press [ZOOM 3]. This will increase the size of the window. If you press [ENTER] again, the window will increase again. If you do this twice, you will notice that the axes look thick and that the graph is hard to see. This is because the tick marks on the axes are set in 1’s. Press [WINDOW] and scroll down to Xscl. If you press [DELETE], this will remove all tick marks. (You can also set the scale to something larger.). To see the graph better, you can also reduce the Xmin and Xmax. Set Xmin to and Xmax to . Press [GRAPH]. Now you can see the function. Press [TRACE] in either direction, and you will be able to see that the left-hand side of the graph continues going down, and the right-hand side continues going up.

  c. The intercepts: to return the graph to a smaller window, press [ZOOM 6]. If you want to see the graph in a smaller window, press [ZOOM 4]. You should see that the graph has intercepts. You can visually approximate them by tracing: press [TRACE] and move the cursor left. The leftmost intercept is around . To find a good approximation of the intercept, press [TRACE 2]. This sends you back to the graph. On the screen you will see the question “Left bound?” Move the cursor to the left of the intercept. (You will be moving down, in this case.) Press [ENTER]. Then you will see the question “Right bound?” Move the cursor to the right of the intercept, but don’t go too far (You don’t want to pass the next intercept.) Press [ENTER]. Then you will be asked to “guess” the intercept. Move the cursor back to the left, as close to the intercept as possible. Press [ENTER]. You should see . This is an approximation of the intercept. If you use the use same steps, you will find that the other intercepts are approximately and .

  d. Maxima and minima: notice that the graph as a “hill” and a “valley.” The hill is called a “local maximum” because it is the highest point on the graph, within a certain interval. The valley is similarly a “local minimum.” To approximate the coordinates of the maximum, press [TRACE] and trace close to the maximum. It appears that the maximum is . To verify this, press [TRACE 4]. To find the maximum, we have to do the same “left bound, right bound, guess” process we used to find the intercepts. This process should tell you that the maximum is . (Note: the value may say something like “” This is just a small calculator error. This number if very close to !) To find the minimum, trace towards the “valley.” (If you want, you can go to the [WINDOW] and make the Ymin a lower number, so that you can clearly see the minimum of the graph.) Now press [TRACE 3]. This will bring you back to the graph. Doing the “left bound, right bound, guess” process should show you that the minimum point is .

  Example 6: You have of fence with which to enclose a plot of land on the side of a barn. You want the enclosed land to be a rectangle.

  a. Write a function to model the area of the plot as a function of the width of the plot.

  b. Graph the function using a graphing calculator.

  c. What size rectangle should you make with the fence in order to maximize the area of the rectangular enclosure?

  d. Explain the significance of the intercepts

  Solution: The plot of land will look like the picture below:

  a. The equation: The area of the rectangular plot is the product of its length and width. We can write the area as a function of : . We can eliminate from the equation if we consider that we have of fence, and we write an equation about how we are using that of fence: . (The fourth side of the rectangle does not require fence because of the barn.) We can solve this equation for and substitute into the area equation:

  b. The graph: Press Y= and clear any equations. Then enter the equation in Y1. Notice that if you press [ZOOM 6], you will not see any graph. You can zoom out by pressing [ZOOM 3], but it may be more efficient to choose a window based on function values. Press [WINDOW] in order to set up the table. TblStart is the first entry you want to see in the table. Tbl allows you to set the increments. For example, if you want to see etc, set this to . For this example, set this to . Make sure Indpnt and Depend ( and ) are set to “auto.” Then press [GRAPH] to see the table. If you scroll through the table, you will see that the value reaches at , and then the values decrease. Now we can set the window. Press [WINDOW]. and set , , , and . (Note: you can set Xmin and Ymin each to , but setting them at and allows you to see the axes.)

  The graph of is shown here on the interval .

  c. The maximum possible area: using the process from example 6, you should find that is the maximum point. This tells us that when the rectangle’s width is , the area is .

  d. Intercepts: Using the process from example 6, you should find that the intercepts are at and . This tells us that if the width of the garden is , then the area is . If the width of the plot of land is , then the area is . This is the case because there is only of fence. If the width is , there is no more fence for the rest of rectangle!

  Now we can return to the weather example.

  Introduction to trigonometric functions

  Consider again the temperature data from above:

  As was noted above, this kind of data needs to be modeled with a function that is periodic. In particular, this kind of data is often modeled by a sinusoid, a graph that oscillates in a particular way, as seen in the graph below.

  Every sinusoid repeats its values on a regular interval. If we modeled the weather data with such a graph, the values will repeat every 12 months. Therefore we say that the period of the function is .

  Notice that the data ranges from about to . Also notice that the “wave” centers in between these values, around . Therefore we say that the amplitude of the wave is about , which is the distance from the middle to the top or the bottom of the wave.

  Many real phenomena can be modeled with this kind of function.

  Lesson Summary

  In this lesson we have reviewed the concept of a function, including major aspects of functions, and different types of functions. We have also used graphing calculators to graph and explore different functions. A key point of this lesson is that we can use functions to model real phenomena. A second key point is that in order to model phenomena that are cyclical in nature, we need to use functions that are periodic. In lesson 4 in this chapter we will define six trigonometric functions. However, because the inputs of these functions are angles, in the next two lessons we will focus on angles. First we will review angles in triangles from Geometry, and then we will consider angles in rotation.

  Points to Consider

  What distinguishes a function from a relation?

  What makes a function periodic?

  What are the pros and cons of using a calculator to graph a function?

  Review Questions

  Determine if each relation is a function:

  A train travels at a constant speed of per hour. Write an equation that shows the relationship between the number of hours the train has traveled and the distance it has traveled.

  Is this situation direct variation, inverse variation, or neither?

  Use the equation to determine the distance the train has traveled after .

  You decide to start a small business making picture frames. You spend
on paint and other supplies, as well as per wooden frame. You decide to sell each frame for . Write a linear function that models the costs of your business

  Write a linear function that models the revenue of your business. (Revenue is the amount of money you take in.)

  Write a linear function that models the profits of your business. (The profits can be found by subtracting the costs from the revenue.)

  Use your profit function to determine the minimum number of frames that must be sold to make a profit.

  Consider the function defined by the equation . To what family does this function belong?

  State the domain and range of the function.

  Use a graphing calculator to graph the function, to identify the approximate coordinates of the vertex, and the approximate values of the intercepts.

  Consider the function Use a graphing calculator to graph the function.

  Identify all asymptotes.

  The price of reserving a private party room in a restaurant is . The price per person varies inversely with the number of people who attend the party. Write an equation that represents the relationship between , the cost per person, and the number of people attending.

  Use the equation to find the cost per person if people attend.

  Use a graphing calculator to graph the functions , , , and . What is the effect of changing the coefficient on the second term?

  The equation represents the profits of a company, where is the number of units the company sells. Use a graphing calculator to graph the function, and use the graph to answer the questions. What is the maximum profit, and how many units must be sold to reach the maximum profit?

  Find the intercepts and explain the meaning of these points on the graph in terms of the profits of the company.

  The table below shows the average daylight hours each month in Anchorage, Alaska. Use your graphing calculator to plot the data, or graph by hand. Use January .

  What is the period of the data?

  How might the graph look different if the data represented daylight hours where you live?

  Month Average daylight hours

  Jan

  Feb

  March

  April

  May

  June

  July

  August

  September

  October

  November

  December

  Review Answers

  Not a function

  Is a function

  Not a function

  The situation is direct variation.

  You must make and sell frames to make a profit.

  This is a quadratic function.

  The domain is the set of all real numbers. The range is the set of all real numbers greater than or equal to .

  Vertex: ; intercepts: .

  The equations with positive coefficients look more and more like , as the coefficient gets larger. The equations with negative coefficients have local maxes and mins. Decreasing the coefficient increases the size of the “hill” and the “valley.”

  Maximum profit is , with sold.

  and . These are the break-even points. When are sold, the company has made enough money to make up for initial costs. After selling , the company is no longer profitable.

  .

  In other U.S. cities, the daylight hours do not vary so greatly. The amplitude of the graph would be smaller.

  Vocabulary

  Dependent variable

  The input variable of a function, usually denoted .

  Domain

  The domain is the set of input values for which a function is defined.

  Function

  A relation in which every element of the domain is paired with exactly one element of the range.

  Independent variable

  The output variable of a function, usually denoted .

  Periodic Function

  Any function that repeats regularly.

  Range

  The set of output or function values for a function.

  Relation

  A pairing between the items in two sets of numbers or data.

  Angles in Triangles

  Learning objectives

  A student will be able to:

  Categorize triangles by their sides and angles.

  Determine the measures of angles in triangles using the triangle angle sum.

  Determine whether or not triangles are similar.

  Solve problems using similar triangles.

  Introduction

  The word trigonometry derives from two Greek words meaning triangle and measure. As you will learn throughout this chapter, trigonometry involves the measurement of angles, both in triangles, and in rotation (e.g, like the hands of a clock.) Given the important of angles in the study of trigonometry, in this lesson we will review some important aspects of triangles and their angles. We’ll begin by categorizing different kinds of triangles.

  Triangles and their interior angles

  Formally, a triangle is defined as a sided polygon. This means that a triangle has sides, all of which are (straight) line segments. We can categorize triangles either by their sides, or by their angles. The table below summarizes the different types of triangles.

  Name Description Note

  Equilateral/equi-angular A triangle with three equal sides and congruent angles This type of triangle is acute.

  Isosceles A triangle with equal sides and two equal angles An equilateral triangle is also isosceles.

  Scalene A triangle with no pairs of equal sides

  Right A triangle with one angle It is not possible for a triangle to have more than one angle (see below.)

  Acute A triangle in which all angles measure less than

  Obtuse A triangle in which one angle is greater than It is not possible for a triangle to have more than one obtuse angle (see below.)

  In the following example, we will categorize specific triangles.

  Example 1: Determine which category best describes the triangle:

  a. A triangle with side lengths and

  b. A triangle with side lengths and

  c. A triangle with side lengths and

  Solution:

  a. This is a scalene triangle.

  b. This is an equilateral, or equiangular triangle. It is also acute.

  c. This is a scalene triangle, but it is also a right triangle.

  While there are different types of triangles, all triangles have one thing in common: the sum of the interior angles in a triangle is always . You can see why this true if you remember that a straight line forms a “straight angle,” which measures . Now consider the diagram below, which shows the triangle , and a line drawn through vertex , parallel to side . Below the figure is a proof of the triangle angle sum.

  If we consider sides and as transversals between the parallel lines, then we can see that angle and angle are alternate interior angles.

  Similarly, angle and angle are alternate interior angles.

  Therefore angle and angle are congruent, and angle and angle are congruent.

  Now note that angles and form a straight line. Therefore the sum of the three angles is .

  We can complete the proof using substitution:

  We can use this result to determine the measure of the angles of a triangle. In particular, if we know the measures of two angles, we can always find the third.

  Example 3: Find the measures of the missing angles.

  a. A triangle has two angles that measures and .

  b. A right triangle has an angle that measures .

  c. An isosceles triangle has an angle that measures .

  Solution:

  a.

  b.

  The triangle is a right triangle, which means that one angle measures .

  So we have: .

  c. and , or and

  There are two possibilities. First, if a second angle measures , then the third angle measures as .

  In the second case, the angle is not one of the congrue
nt angles. In this case, the sum of the other two angles is . Therefore the two angles each measure .

  Notice that information about the angles of a triangle does not tell us the lengths of the sides. For example, two triangles could have the same three angles, but the triangles are not congruent. That is, the corresponding sides and the corresponding angles do not have the same measures. However, these two triangles will be similar. Next we define similarity and discuss the criteria for triangles to be similar.

  Similar triangles

  Consider the situation in which two triangles have three pair of congruent angles.

  These triangles are similar. This means that the corresponding angles are congruent, and the corresponding sides are proportional. In the triangles shown above, we have the following:

  Three pair of congruent angles: , and

  The ratios of sides within one triangle are equal to the ratios of sides within the second triangle: , and

  The ratios of corresponding sides are equal: , and

  Example 4: In the triangles shown above, , and . What are the lengths of sides and ?

  Solution: and .

  Given that , we have .