Ohio Standards Alignment
- Angles in radian measure
- Definitions of trigonometric ratios
Review of right triangle trigonometry, unit circle definition of trigonometric functions, radian measure, graphs of trigonometric functions
Students compare trigonometric ratios for a unit and non-unit circle with the goal of observing that the ratios are independent of the radius. They thus see the benefit of using a radius of one unit. Using the unit circle, students generate a trig table for angles in Quadrants I and II by connecting the x-coordinate to the cosine ratio and the y-coordinate to the sine ratio. Through pattern recognition the table is extended to Quadrants III and IV. The table is used to graph the three basic trigonometric functions -- sine, cosine, and tangent. This problem is appropriate for use after learning radian measure and as an opening activity to the unit circle.
Looking at the diagram of the unit circle, angle DGO is a right angle. The radius of the circle is 1 unit. With a protractor measure angle DOG. Label the measures of segments OG, DG and OD. Find the sine ratio of angle DOG. Find the cosine ratio of angle DOG.
What do you notice about the sine and cosine of angle DOG?
Distribute the student worksheet. Ask students to complete Questions 1, 2, 3 and 4. A whole-class discussion should follow from Questions 3 and 4. Students should observe, when using the unit circle, the sine ratio is the value of the y-coordinate and the cosine ratio is the value of the x-coordinate (as the denominators have a value of 1). Moreover, the measure of the intercepted arc is the radian measure of the angle.
Students should then complete Questions 5 and 6. Have students discuss conjectures about patterns found in the table.
What is the advantage of using a circle that has a radius of 1 unit?
Without the benefit of drawing the angles on the unit circle, students should use observed patterns to complete Question 7.
Is the sine ratio always defined? Is the cosine ratio always defined? Is the tangent ratio always defined?
Questions 8 - 15 are follow-ups that lead to the graphs of the three basic trig functions.
In which quadrants is the sine ratio negative? In which quadrants is the cosine ratio negative? In which quadrants is the tangent ratio negative?
What is the domain for each of the 3 trig functions? What is the range for each of the 3 trig functions? Which graphs are continuous and why?
When looking for patterns, prompts may be given such as: Find a relationship between the sine and cosine ratios. Is the rate of change for the individual ratios constant? When are the ratios increasing, decreasing, etc?
Students working in groups can average the ratios they find to get better estimates.
A discussion point can be that quadrantal angles do not create right triangles, but their cosine and sine ratios can be defined from the x and y-coordinates.
Students may need some help understanding that, while the angles are defined relative to the x-axis, the measurement of all angles is from the positive x-axis counterclockwise, thus allowing for angle measurements over 90°.
Students can check the accuracy of ratios using their calculators and also the graphs they obtain.
From the teaching files of Teresa Graham.