Cogito
Trigonometry · Chapter 5 · Lesson 2
Proving an Identity
Work one side until it turns into the other.
12 problems · about 22 minutes · F-TF.C.8
What this lesson teaches
The student verifies trigonometric identities using the Pythagorean, reciprocal and quotient identities.
- An identity holds for every angle.
- Work one side only, starting with the messier one.
- Rewriting everything as sine and cosine almost always opens the way.
Warm Up
Straightforward practice. Get the method working first.
5 problemsWhat is sin²θ + cos²θ for any angle?
Answer 1
Why 1.
Why may you not move terms across an identity you are proving?
Answer It assumes the equation is true, which is the thing being proved.
Why It assumes the very statement you are trying to prove.
sin²θ = 0.49. What is cos²θ?
Answer 0.51
Why 1 − 0.49.
tan²θ + 1 = 9. What is sec²θ?
Answer 9
Why They are the same quantity.
cot²θ = 15. What is csc²θ?
Answer 16
Why 1 + cot²θ.
Build It Up
The same ideas with more to keep track of.
3 problemsMatch each expression with its simplest form.
Answer sin θ · csc θ → 1; sin θ ÷ cos θ → tan θ; 1 − cos²θ → sin²θ
Why A ratio times its own reciprocal is 1.
cos θ · sec θ equals what number?
Answer 1
Why A ratio times its reciprocal.
sec²θ − tan²θ equals what number?
Answer 1
Why Rearrange tan²θ + 1 = sec²θ.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the proof of tan θ · cos θ = sin θ in order.
Answer 1. Start with the left side, tan θ · cos θ. 2. Replace tan θ with sin θ ÷ cos θ. 3. Cancel the cos θ terms. 4. The left side is now sin θ, matching the right.
Why Rewriting comes before cancelling.
cos θ = 0.8 and θ is acute. What is sin θ?
Answer 0.6
Why sin²θ = 1 − 0.64.
The Substitution: If sin²θ = 0.36, what is cos²θ?
Answer 0.64
Why 0.64.
The First Move: To prove sec θ · cot θ = csc θ, what is the best opening move?
Answer Rewrite the left side as sine and cosine.
Why Rewrite one side in sine and cosine. Touching both sides assumes the result.