Section 7.3: Investigation of Circular Functions

Honors Analysis

Section 7.3: Investigation of Circular Functions

Joaquin is enjoying the amazing spring weather by taking his bike out for a spin. As he basks in the first rays of the springtime sun, he unwittingly rides over a freshly-painted stripe on the road, resulting in a spot of yellow paint on his front wheel that spins at the same rate as his wheel. Likewise, he is blissfully unaware of the wealth of mathematics to be observed from his mistake.

In the following activity you will analyze the motion of his wheel by graphing, and as a result, you will uncover many new relationships in the world of trigonometry.

Graphing: Degrees & Position

To analyze the relationships imbedded in circular motion, we will superimpose the tire on a coordinate plane with the center at the origin. Assume the tire has a 24 inch diameter.

1.)In the table below, you will plot the height of the paint spot above and below the origin. Plot points below the x-axis as negative heights. Keep in mind that due to the symmetry of the tire, you may not need to calculate every height from scratch.

Assume that the wheel is being spun in a positive (counter-clockwise) direction and that the paint spot begins on the far right side of the wheel.

Degrees Tire Has Rotated / 0º / 30º / 45º / 60º / 90º / 120º / 135º / 150º / 180º
Height Above & Below X-Axis
(Exact form AND Decimal)
Degrees Tire Has Rotated / 210º / 225º / 240º / 270º / 300º / 315º / 330º / 360º
Height Above & Below X-Axis
(Exact AND
Decimal)

2.)Now plot the results on the graph below:

3.) Continue your graph for angles of rotation up to 540º and back to -180º. How should you be able to do this without calculating each value?

4.) Next you are going to graph the sine function. Use a unit circle (centered at the origin, with a radius of one unit) to fill in the table below for the function y = sin x.

X / 0º / 30º / 45º / 60º / 90º / 120º / 135º / 150º / 180º
Y (Exact form AND as decimal)
X / 210º / 225º / 240º / 270º / 300º / 315º / 330º / 360º
Y (Exact form AND as decimal)

5.) Plot the values from your table to create the graph of the function y = sin x. Then extend the pattern back to -180º and forward to 540º.

6.) How is the graph above similar to the graph of the wheel motion? How is it different?

7.) What is the amplitude of the sine function? The period?

8.) Next you are going to graph the cosine function. Use a unit circle to fill in the table below for the function y = cos x.

9.) Use your table to graph the function y = cos x below:

10.) What is the amplitude? What is the period?

11.) Compare and contrast the graphs of the sine and cosine functions.

Transformations of Trig Functions

12.) Graph the function y = sin x on your graphing calculator. You may want to use the interval [-180, 540] for the x-axis and [-5, 5] for the y-axis.

13.) Graph each at the same time as the sine function. Explain how the function is transformed in comparison to the parent function y = sin x.

A)y = 3 sin xB) y = -3 sin xC) y = sin x + 1D) y = sin x – 2E) y = sin (x + 30)F) y = sin (x – 30)G)y = sin (2x)H)

14.) Without graphing, explain how y = 4 sin (x – 60) + 2 might change compared to y = sin x. Check your prediction – were you correct?

15.) Graph each at the same time as the cosine function. Explain how the function is transformed in comparison to the parent function y = cos x.

A)y = 4cos xB) y = -4cos xC) y = cos x + 3D) y = cos x – 4E) y = cos (x + 60)F) y = cos (x – 60)G) y = cos (3x)H)

16.) Use your observations from #12-15 to write a sine or cosine function matching the equation you graphed for the bicycle problem. Check your answer on the graphing calculator.