Cogito
Integrated Math 3 · Chapter 6 · Lesson 1
The Pythagorean Identity
One equation the circle guarantees.
12 problems · about 21 minutes · F-TF.C.8
What this lesson teaches
The student applies the Pythagorean identity and its rearrangements.
- sin²θ + cos²θ = 1 holds for every angle.
- It is the unit circle equation with the coordinates renamed.
- Dividing by cos²θ or sin²θ gives the other two forms.
Warm Up
Straightforward practice. Get the method working first.
5 problemssin²θ = 0.64. What is cos²θ?
Answer 0.36
Why 0.36.
Where does the Pythagorean identity come from?
Answer The unit circle equation x² + y² = 1.
Why It is the circle equation renamed.
sin²θ = 0.49. What is cos²θ?
Answer 0.51
Why 1 − 0.49.
cos θ = 0.8 with θ acute. What is sin θ?
Answer 0.6
Why sin²θ = 1 − 0.64.
What is sin²θ + cos²θ for any angle?
Answer 1
Why The identity.
Build It Up
The same ideas with more to keep track of.
3 problemstan²θ + 1 = 9. What is sec²θ?
Answer 9
Why They are the same quantity.
cot²θ = 15. What is csc²θ?
Answer 16
Why 1 + cot²θ.
sec²θ − tan²θ equals what number?
Answer 1
Why Rearrange the identity.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsMatch each expression with its simplest form.
Answer One minus cos squared theta → sin squared theta; One minus sin squared theta → cos squared theta; sin theta over cos theta → tan theta
Why The identity handles the first two.
sin²θ = 0.25. What is cos²θ?
Answer 0.75
Why 1 − 0.25.
The Square: sin²θ = 0.36. What is cos²θ?
Answer 0.64
Why 0.64.
The Ratio: sin θ = 0.6 with θ acute. What is cos θ?
Answer 0.8
Why 0.8.