Even, Odd, and Cofunction Identities

What symmetry gives you for free.

12 problems · about 20 minutes · F-TF.C.8, F-TF.A.4

What this lesson teaches

The student applies even-odd and cofunction identities to rewrite trigonometric expressions.

Cosine is even; sine and tangent are odd.
A cofunction identity swaps a function for its co-partner at the complementary angle.
Both rules come straight from symmetry on the unit circle.

Straightforward practice. Get the method working first.

5 problems

sin θ = 0.9. What is sin(−θ)?
Answer -0.9
Why −0.9.
Which function is even?
Answer Cosine.
Why Cosine, because it follows x, which the reflection leaves alone.
cos θ = 0.7. What is cos(−θ)?
Answer 0.7
Why Cosine is even.
tan θ = 3. What is tan(−θ)?
Answer -3
Why Tangent is odd.
cos 20° ≈ 0.94. What is sin 70°?
Answer 0.94
Why 20 and 70 total 90.

The same ideas with more to keep track of.

3 problems

sin(−30°) as a decimal?
Answer -0.5
Why Sine is odd, and sin 30° = 0.5.
cos(−60°) as a decimal?
Answer 0.5
Why Cosine is even.
If sin θ = cos 40°, what is θ in degrees?
Answer 50
Why They must be complementary.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Which of these functions are odd?
Answer sine; tangent
Why An odd function flips sign when the input does, and cosine does not.
tan(−45°) as a decimal?
Answer -1
Why tan 45° = 1, and tangent is odd.
The Flip: sin θ = 0.4. What is sin(−θ)?
Answer -0.4
Why −0.4.
The Complement: sin 25° ≈ 0.423. What is cos 65°?
Answer 0.423
Why About 0.423, by the cofunction identity.