Trigonometry and the Unit Circle

Lesson One: Angles and Angle Measures

Specific Outcome 1: Demonstrate an understanding of angles in standard position, expressed in degrees and radians.
Achievement Indicators:
1.1 Sketch, in standard position, an angle (positive or negative) when the measure is given in degrees.
1.2 Describe the relationship among different systems of angle measurement, with emphasis on radians and degrees.
1.3 Sketch, in standard position, an angle with a measure of 1 radian.
1.4 Sketch, in standard position, an angle with a measure expressed in the form kπ radians, where k∈Q.
1.5 Express the measure of an angle in radians (exact value or decimal approximation), given its measure in degrees.
1.6 Express the measure of an angle in degrees, given its measure in radians (exact value or decimal approximation).
1.7 Determine the measures, in degrees or radians, of all angles in a given domain that are coterminal with a given angle in standard position.
1.8 Determine the general form of the measures, in degrees or radians, of all angles that are coterminal with a given angle in standard position.
1.9 Explain the relationship between the radian measure of an angle in standard position and the length of the arc cut on a circle of radius r, and solve problems based upon that relationship.

Look at the explanation of Trigonometry on p. 162. Show the students how trig has real life examples.

Need:
Compass
Pipe cleaners
Protractor

Working with a partner:

1. Draw circles with diameters of 10cm, 14cm, and 20cm (be sure to leave plenty of space between circles). Label the radius of each circle as CA.

2. Cut a pipe cleaner to a length equal to radius CA for each circle.

3. Place and bend the pipe cleaner (the one from step 2 above) around the circumference of your circle, beginning at point A and ending at point P, as shown by the arrow arc below:

4. On the circle you are working with, draw radius CP.

5. Measure ∠PCA with a protractor. What is the measure of ∠PCA in degrees? Is the measure of ∠PCA the same for all three circles?

Should be around 60˚; all circles will have the same result.

6. Use the pipe cleaner to continue marking radius lengths all the way around your circle. About how many radius lengths are there in one complete circumference of your circle? Should be around 6 lengths.

7. The circumference, C, of a circle is given by the formula C = 2πr where r represents the radius. Explain how this formula relates to your answer for part F.

For example, take the circle with radius of 5. It has 6 equal radius lengths for its circumference, therefore the circumference is equal to If you solve for the circumference using the formula, you get .

8. What does the circumference formula find for us? What degree measure is equal to 2π radians (see definition below)?

Circumference finds the distance around a circular object, therefore 2π is equal to 360˚.

The measure of ∠PCA is considered to be 1 radian. Approximately 60˚, closer to 57.3˚.

Radian: the measure of the central angle (∠PCA) of a circle subtended by an arc that is the same length as the radius of the circle.

What degree benchmark can we use for 1 radian?

60˚ (more like 57.3), 1 radian is about

One radian is the measure of the angle at the centre of a circle subtended by an arc equal in length to the radius of the circle. This brings us the formula: .

We can measure angles, therefore, in degrees and radians.
One full rotation is 360˚ or 2π radians. One half rotation is 180˚ or π radians
One quarter rotation is 90˚ or radians. One eight rotation is 45˚ or radians

Angle measures without units are considered to be in radians.

Conversion Chart
Degrees to Radians multiply by Radians to Degrees multiply by

You will need to be able to convert back and forth from degrees to radians and vice versa. We will also be looking at putting angles on coordinate planes.

Do you remember what an angle