Instructional Unit

Trigonometric Functions

I made a instructional unite about trigonometric functions. I start with the definition of trigonometry functions. Our final goal is to show the graphs of sin(x), cos(x), and tan(x) by means of GSP. In my teaching unite, there are several exercises that students learn as their home work or their classroom assessment.

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Day1, 2, 3, and 4

Introduction and sone basic ideas of trigonometric functions with a right triangle

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Day 5, 6, 7, and 8

Trigonometric functions with a unite circle

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Day 9 and 10

The graphs of sin(x), cos(x), and tan(x) shown by GSP.

Day1

Investigation of similar triangles

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<Examples>

Two triangle ABC and A'B'C' are similar. Think about the ratios of each two sides of a right triangle.

Question: What are AC/AB and A'C'/A'B'? What are BC/AB and B'C'/A'B'? What are BC/AC and B'C'/A'C'? Compare the result.

Result:

The two similar triangles have the same ratios of two sides.

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Try on another two right triangles.

Think about the ratios of each two sides of a right triangle.

Question: What are DF/DE and D'F'/D'E'? What are EF/DE and E'F'/D'E'? What are EF/DF and E'F'/D'F'? Compare the result.

Result:

The two similar triangles have the same ratios of two sides.

We observe that similar right triangles have the same ratios of each two sides. Another thing we got as a result is that if two right triangles are not similar, they have different ratios of each two sides.

Here we start thinking of the relationship between the angle and the ratios of two sides of a right triangle.

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Definition of trigonometric functions

Exercise

Calculate and compare the values of sin, cos, and tanfor both triangle below.

Day2

Trigonometic values with various angles

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In this day we start over with an exercise problem in order to review and prepare for intrducing new concept.

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Exercise

Calculate the values of sin, cos, and tan, where = 30, 45, 60

Think about this problems for some right triangles with various lengths.

The answer of this problems are below.

Different angles give different values of sin, cos, and tan. sin, cos, and tan depends on an angle.

We are familiarized with the right triangle of angles 30, 40, and 60. How about a triangle with other angles between 0and 90? It is hard to find values for most other angles. By spread sheet, we can look at how sin, cos, and tan are changing to . The values shown in the cells are apploximation of the values.

Think of each value of sin, cos, and tanwhen= 90 or 0. We need to extend the definition of trigonometric function for considering the case that is not between 0 and 90.

Exercise

What is sin63?

What is cos19?

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Exercise

What is an angle A so that sinA=0.39?

What is an angle A so that cosA=0.53?

Day3

Application of sin, cos, tan.

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Why people made the trigonometric functions? How useful are they? One obvious application of them to the real world is measurement of some length. Consider the problem below.

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A helicopter hovers 800 ft directly above a small island that is off the California coalst. From the helicopter, the pilot takes a sighting to a point P directly ashore on the mainland, at the water's edge. If the angle of depression is 35, how far off the coast is the island?

Solution

Let x denote the distance from the island to the mainland. Then, as you can see from Figure 7, we have +35=90, from which it follows that =55. Now we can write

tan55=x/800

or

x= 800 tan551150 ft

(using a calculator and rounding to the nearest 50 feet)

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Exercise

Determine the distance AB across the lake shouwn in the figure, using the following data: AC=400m, C=90, and CAB=40.

Day4

Relationship between trigonometric values shown with a right triangle.

By the difinition, define each trigonometry value for two angles and 90- of the triangle above and compare the values.

Result:

Thus, we got below relationship between one with and one with 90-.

Exercise

Use an angle smaller than 45 to express each below

sin75

cos65

tan 72

Day5

(cos, sin)on the unite circle

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We need to give new definitions of sin and cos to think of the case of >90.

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Step1

Think of a right triagnle, review the difiniton of sinand cos.

Step2

If the length z is equal to 1, sinis equal to y and cosis equal to x like below.

Step3

Consider the point with interior angle is on the origin point on xy plane. Then, since the two smallest sides are sinand cos, the point P in above picture has a coordinate (sin, cos).

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Step4

The coordinates of point P changes as the anglechanges between 0and 90.

It seems that the point P is on a circle.

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Step5

Even if is greater than 90, we can define the value of cosand sin as the coordinates of the point P.

The angle can be any angle between 0and 360(actually, any big angle greater than 360 or any negative angle), Whatever is, the value of sin and cos is the coordinates of the point P.

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Excercise

find the value of sin and cos for each angle below.

=0, 90, 120, 180, 225, 300.

Day6

Relationship between trigonometric values shown with the unite circle (1)

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Let us continue to deal with value of sin and cos with many different angles of . Here is a work sheet for students to fill the value of trigonometric funcion for each angle between 0 and 360.

0 | 30 | 45 | 60 | 90 | 120 | 135 | 150 | 180 | 210 | 225 | 240 | 270 | 300 | 315 | 330 | 360 | |

sin | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ |

cos | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ |

@ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ |

It is not important to fill all the cell of this table but to think about what is in each of the very bottom cells. While filling this table, especially the very bottom cells, we may realize in a short time that . The reason of equation is very obvious. It is because that cosand sinare each of the coordinates of a point on the circle with its center of the origin, that is (0, 0) and with its radius of 1.

It is easy to see this with spread sheet like below.

Can you find any other relationship between them? How about tan?

Actually, tan has a property expressed with sin and coslike below.

The proof of this is not so easy but not so complicated eather. Think about a right triangle again.

Usining this result(), we can obtain another equation of tanand cos, starting on.

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Exercise

When cos=-3/4 and 0<<180, find sin and tan.

Day7

Relationship between trigonometric values shown with the unite circle (2)

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Now we can discuss trigonometric functions with an angle greater than 90 and we know that

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Let us see what happens if we change the minus sigh into plus sign. I mean, think about.

In this picture, the point A and the point A' has an angle and 90+ respectively. Thus, the coordinates of A is (cos,sin) and the coordinates of A' is (cos(90+), sin(90+)). Because the triangle AOB and the triangle A'OB' are similar, the each pair of the sides colored blue and red has the same length.

Finally, paying much attention to signs, we get

Next, I need to deal with tan(90+). Using the result we have got, we can change tan(90+) into a simple form with , like below.

Here is a GSP file I made to show this trigonometric relationship.

Exercise

Consider the followings. Express them only with .

sin(180+)

cos(180+)

tan(180+)

sin(180-)

cos(180-)

tan(180-)

sin(270+)

cos(270+)

tan(270+)

sin(270-)

cos(270-)

tan(270-)

sin(360+)

cos(360+)

tan(360+)

sin(360-)

cos(360-)

tan(360-)

Day8

Value of tanaround the unite circle

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A pair (cos, sin) is a coordinates of a point on the unite circle. How about tan? Where is it around the unite circle? Think about this by starting over an right triangle again.

If the length x is equal to 1, then the length y is equal to the length y.

Thus, we have found the value of tan within the unite circle, as the y-coordinate of the point P on the line that is x=1. Note that the base of the triangle is 1.

Even when the angle is more than 90, tanis defined as the y-coordinate of the pointP on the line x=1

Note: The length of the thick segment is the absolute value of tan.

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Exercise

Make it sure that is correct when 0<<360.

Day9

The graph of y=sin(x)

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In this part of our lessen, we show by a GSP file (Click here to download this file) students how to draw the graph of sin(x) .

In the file, there are three bottons,

(I)Rolling movement, (II)Horizontal movement, and (I)+(II).

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If you push the button (I), the point A moves around on the unite circle and the point P also moves on y-axis along with the point A.

Stop the movement of button (I). Then, if you push the button (II), the point P moves right horizontally.

Go back to the first position (and erace traces). If you push the third button (I)+(II), the point A moves around and the point P moves horizontally at the same time. As a result, a curve like a wave appers. This curve is a graph of y=sin(x).

AOD is transformed into the value of x in y=sin(x).

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Exercise

Change the starting position of the point A, P, or both. Try various positions of the points and interprete the curve of a result.

Consider an equation y= sin(ax+b). How does this graph change if a and b changes independently.

Day 10

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The graph of y=cos(x)

Because cos=sin(+90), the graph of y=cos(x)(=sin(x+90)) starts at the point with the coordinates (0, 1). If you push the button (I), the point Q goes down on the y-axis as the point B moves around on the unite circle.

IF you push the button (I)+(II), the point Q leads a curve, which is the graph of y=cos(x).

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The graph of y=tan(x)

The value of tan(x) is the y-coordinate of the point C on the line x=1. The point R goes up along with the point C as the point A moves around on the unite circle.

If you push the button (I)+(II), the graph of y=tan(x) appears behind the point R.

We can see that when is 90, 270, 450, ... the value of tan(x) does not exist.

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Exercise

Where each two graphs of the tree intersect? What is the angle giving the intersections? Why?