Find exact values for the sine and cosine of angles not on the unit circle by writing the angle as a sum or difference of known angles.
Use equivalent trigonometric expressions arising from the angle sum and double angle identities to solve equations.
Quick Lesson Plan
Today we’ll be using another diagram to help us see the angle sum identities. We chose not to go very in depth about how the diagram was created and why, but if you have extra time this would be a very rich and worthwhile discussion to have with your students. For example, why does angle alpha appear in three places??
In question 2 students can reason visually why the sine of the sum of the two angles is not equal to the sum of the sine of each angle. If students are stuck, you can suggest students try substituting specific values like alpha=π/3 and beta=π/6 since the sum will be π/2, or any two angles, like π/4 and π/4.
In question 3, students write expressions for sine and cosine of alpha and beta and then in questions 4-6 we let them wrestle through how they might combine those expressions to find a value for DF and FE which make up DE. Note that instead of finding FE directly, they will find GH which is a congruent segment and thus has equal length.
Page 2 is a parallel of the first place except now they think about the cosine of two angles. Note here that the segment whose length represents the cosine (segment AE) is the difference between segment HA and segment EH. This makes the argument in question 9 fairly easy to see. Adding two segments that are both longer than AH will certainly yield a greater result than subtracting a positive value from the smaller segment AH. Note that the inequality is true in the first quadrant, but it is almost always true that the cosine of the sum is NOT equal to the sum of the cosine of each angle.
There are a few ways you may wish to modify the lesson based on the needs of your students.
Do the first page as a whole class, then have students work in groups on page 2 (most scaffolded).
Have students work on the first page in groups but debrief page 1 before having students move on to page 2 (moderately scaffolded).
Students do both sides in their groups, then the entire lesson is debriefed (least scaffolded).
Which segment’s length represents sin alpha?
Which segment’s length represents sin beta?
Can you rewrite DE in terms of DF and GH?
Which expressions do you think will be helpful for finding DF? Why?
Which expressions do you think will be helpful for finding FE (GH)? Why?
What do you think would change if we subtracted two angles (or added a negative angle)?
When you are debriefing the lesson, make sure you talk about how the identity for a difference of two angles is very closely related to the identity for the sum of two angles. In the Check Your Understanding, students will extend the angle sum identity to create the double angle formula.
In the QuickNotes, students should start to see some throughlines of the course. In almost every unit we have learned about how to write equivalent forms of functions and discussed why those forms are helpful. The ability to solve a previously unaccessible equation has been a consistent motivation. Continue to emphasize the why behind these algebraic manipulations and substitutions, so they don’t just become rote procedures.
This is a great Check Your Understanding to have students complete in their groups. Note that it may take students some time to determine how the 255˚ angle can be written as a sum or difference of two angles on their unit circle. That’s okay! Let them wrestle through it and talk as a team!