Cogito
Trigonometry · Chapter 5 · Lesson 1
The Pythagorean Identity
Pythagoras, on a circle of radius 1.
10 problems · about 20 minutes · F-TF.C.8
What this lesson teaches
The student states and applies the Pythagorean and quotient identities.
- sin²θ + cos²θ = 1 for every angle.
- It is Pythagoras applied to the unit circle.
- tan θ = sin θ ÷ cos θ.
Warm-Up Review
From earlier lessons. Loosen up before the new work.
2 problemsReview — Composing Functions and Inverses: What is arcsin(sin 170°), in degrees?
Answer 10
Why 10°.
Review — Solving Trigonometric Equations: sin θ = 0.866 gives 60°. The other solution below 360°, in degrees?
Answer 120
Why 120°.
Warm Up
Straightforward practice. Get the method working first.
4 problemssin θ = 0.28 and cos²θ = 1 − 0.0784. What is cos²θ?
Answer 0.9216
Why 0.9216.
Where does sin²θ + cos²θ = 1 come from?
Answer Pythagoras, on a triangle with hypotenuse 1.
Why It is Pythagoras in circle notation.
cos θ = 0.6, θ acute. What is sin θ?
Answer 0.8
Why 1 − 0.36.
sin θ = 0, θ = 0. What is cos θ?
Answer 1
Why 1 − 0.
Build It Up
The same ideas with more to keep track of.
2 problemsWhat is sin²30° + cos²30°?
Answer 1
Why True for every angle.
tan θ equals sin θ divided by what?
Answer cos θ.
Why y over x.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
2 problemsThe Identity: sin θ = 0.8 and θ is acute. What is cos θ?
Answer 0.6
Why 0.6.
The Sum: What is sin²θ + cos²θ, for any θ?
Answer 1
Why 1, always.