Unit 6 Modeling Periodic Behavior

Lesson 1

Learning Focus

Apply right triangle trigonometry to a circular context.

Lesson Summary

In this lesson, we learned how to place reference right triangles on a circle in order to find the distance a point on the circle is above or below the center of the circle. This is useful for finding the height above ground of a point on a circular object like a bicycle tire or a Ferris wheel.

Lesson 2

Learning Focus

Write a trigonometric function to model a context.

Lesson Summary

In this lesson, we found an equation for the vertical height of a rider on a moving Ferris wheel. That is, we treated the height of the rider as a function of the elapsed time since the rider passed the starting position, which we considered to be the farthest right position on the wheel.

Lesson 3

Learning Focus

Extend the definition of sine to include all angles of rotation.

Lesson Summary

In this lesson, we extended the definition of the sine to make it possible to find sine values for nonacute angles, including all possible angles of rotation $- \infty < θ < \infty$ . We also learned how to draw angles in standard position on a coordinate grid, and to decide if angles are coterminal, and therefore have the same value for the sine function.

Lesson 4

Learning Focus

Graph sine functions of the form $h (t) = a \sin (b t) + d$ .

Lesson Summary

In this lesson, we learned how to represent circular motion using a description, an equation, and a graph. We related the parameters $a$ , $b$ , and $d$ in the equation $f (t) = a \sin (b t) + d$ to the description of a Ferris wheel, and to the midline, amplitude, and period of a sine graph.

Lesson 5

Learning Focus

Extend the definition of cosine to include all angles of rotation, and use cosine functions to model a context.

Lesson Summary

In this lesson, we learned how to graph the horizontal position of a rider on the Ferris wheel using the cosine function. This required that we extend the definition of the cosine to include all angles of rotation. We examined attributes of the cosine graph and its equation and related those to similar attributes of the sine function.

Lesson 6

Learning Focus

Locate points in a plane using coordinates based on horizontal and vertical movements or based on circles and angles.

Use degrees and radians to measure angles.

Lesson Summary

In this lesson, we learned how to locate points in a plane using either rectangular or polar coordinates. We also revisited the definition of the radian measurement of an angle.

Lesson 7

Learning Focus

Calculate arc length for angles of rotation measured in radians.

Visualize the size of angles measured in radians, including radians given in decimal form.

Lesson Summary

In this lesson, we continued to work with degree and radian measurement for angles of rotation. We found strategies for converting from one angle measurement to the other, and we saw that the formula for finding arc length for angles measured in radians was simpler than the formula for finding arc length for angles measured in degrees. This occurred because radian measure is defined as a ratio of arc length to radius.

Lesson 8

Learning Focus

Find a relationship between the arc length and $(x, y)$ coordinates of a point on the circle of radius $1$ .

Lesson Summary

In this lesson, we learned about the unit circle and how it models the sine and cosine values for every angle of rotation. We also found that the radian measure of an angle of rotation is represented by the arc length of the intercepted arc.

Lesson 9

Learning Focus

Apply special right triangles to the unit circle.

Lesson Summary

In this lesson, we learned that the values of some trigonometric expressions can be found exactly, instead of as decimal approximations. This occurs because we can find the exact side lengths for special right triangles with a hypotenuse of $1 unit$ . We can then use these lengths to label the coordinates of points around the unit circle that can be identified by placing these right triangles in various positions where they fit within the unit circle.