Extending trigonometric functions to the real numbers
• See coordinates on the unit circle as the sine and cosine of an angle (F-TF.A.2).
• Understand radian measure and convert between radians and degrees.
• Graph basic trigonometric functions using radians as the x-axis scale (F-IF.C.7e$^\star$).
Students have understood trigonometric ratios in terms of right triangles, using them to solve for various sides and angles. In this section they make the big transition to thinking of trigonometric ratios as functions. There are two important steps in this transition. First, students re-envision trigonometric ratios on the unit circle, and then use the idea of an angle as a movement around a circle to extend the definition to angles of any measure. Second, they use the unit circle to understand a new way of measuring angles, radian measure, which uses distance around the circumference of the circle to measure an angle, rather than an arbitrary division of the circle into 360 degrees. From now on students will be thinking of sine and cosine as functions with numerical inputs, in addition to than ratios related to angles.
Tasks
1
F-TF Trigonometric functions for arbitrary angles (degrees)
WHAT: This task guides students through the definition of the sine and cosine of the obtuse angles $135^\circ$ and $180^\circ$ using the unit circle.
WHY: In the previous task students found coordinates of points on the unit circle. Now they see that those coordinates can be interpreted as cosine and sine for any angle. In order to understand why the coordinates of a point on the unit circle can be written $(cos a, sin a)$ and extend this definition to points beyond the first quadrant, students will need to reason abstractly and quantitatively (MP.2).
2
What exactly is a radian?
WHAT: Students are given a printed unit circle with dots indicated by 10 degrees, 20 degrees, etc., all the way around the circle. They are tasked with finding the measure in degrees of angles that measure 0, 1, 2, 3, 4, 5, and 6 radians. They are also asked to estimate the measure in radians of the angles that measure 180 and 360 degrees.
WHY: The purpose of this task is to introduce the idea of radian measure. Now that students have seen the unit circle as a device for extending the definition of trigonometric functions, they can begin to have a different view of angle measure. The unit circle invites thinking about angle as measured by distanced around the perimeter. The task suggests that students to use a ruler or a piece of paper to “bend the radius around the circle,” giving students an opportunity to use appropriate tools and attend about precision (MP.5, MP.6). After students find the approximate radian measure for 180 and 360 degrees, the teacher can point out that the exact corresponding measures in radians are $\pi$ and $2\pi$, respectively.
3
Bicycle Wheel
WHAT: Students consider a bicycle wheel with a radius of 13 inches. The make a table of angle measurements corresponding to various distances around the bicycle wheel, first in degrees and then in radians.
WHY: The purpose of this task is to present a situation where radian measure is the natural choice. The table of radian measures in terms of distance is simpler than the table of degree measures, because π does not appear in the values. The spinning of a bicycle wheel is a natural situation in which to consider the radian measure for angles, because the distance the bicycle travels is measured by length around the perimeter of the wheel. The task can also serve as the launch for a discussion of how to convert between radians and degrees (MP.2).
4
Trig Functions and the Unit Circle
WHAT: Students are presented with two diagrams, one showing the graphs of sine and cosine with some points marked and the other showing a unit circle with a point marked. They are asked to relate the coordinates of the points on the graphs with the coordinates and angle measure of the point on the unit circle.
WHY: The purpose of this task is to bring together the two big concepts that students have learned in this section: extending trigonometric functions using the unit circle and radian measure of angles, and to assess students’ understanding of the connections between these and the graphs of the sine and cosine functions.
External Resources
1
Bring the Problem to Physical Reality: Trigonometry
Description
WHAT: Students are given a large unit circle printed on graph paper and a list of angles. They use the grid (MP.5) to record precise coordinates for the point on the unit circle corresponding to each angle and tabulate the results.
WHY: The purpose of this activity is to introduce students to the correspondence between points on the unit circles and angles, and to observe the changes in the coordinates of the point as the angle changes. They see the coordinates start at $(1,0)$ for $0^\circ$ slowly change to $(0,1)$ as the angle approaches $90^\circ$. They see patterns as they move through the four quadrants (MP.8) that prepare them to understand why certain angles have related trigonometric ratios (F-TF.A.2).