Lesson 4: Even-Odd Properties and Cofunction Identities
Estimated time: 25-30 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Identify which trig functions are even and which are odd
- Apply even-odd properties to simplify expressions involving negative angles
- State and use cofunction identities for all six functions
- Combine identities to simplify complex expressions
Even and Odd Functions Review
Even function: f(−x) = f(x). Symmetric about the y-axis.
Odd function: f(−x) = −f(x). Symmetric about the origin.
Even-Odd Properties of Trig Functions
Even functions: cos(−θ) = cos θ and sec(−θ) = sec θ
Odd functions: sin(−θ) = −sin θ, tan(−θ) = −tan θ, csc(−θ) = −csc θ, cot(−θ) = −cot θ
Why? On the unit circle, replacing θ with −θ reflects the point across the x-axis: (x, y) becomes (x, −y). So x (= cos) stays the same, but y (= sin) changes sign.
Example 1: Applying Even-Odd Properties
Simplify: sin(−60°)
Solution: sin(−60°) = −sin 60° = −√3/2
Example 2: Cosine is Even
Simplify: cos(−π/4)
Solution: cos(−π/4) = cos(π/4) = √2/2 (cosine is even, so the negative sign drops)
Example 3: Simplifying an Expression
Simplify: sin(−θ) + cos(−θ)
Solution: −sin θ + cos θ = cos θ − sin θ
Cofunction Identities
Cofunctions relate trig functions of complementary angles (θ and π/2 − θ):
sin θ = cos(π/2 − θ) cos θ = sin(π/2 − θ)
tan θ = cot(π/2 − θ) cot θ = tan(π/2 − θ)
sec θ = csc(π/2 − θ) csc θ = sec(π/2 − θ)
Example 4: Using Cofunctions
Write cos 35° as a function of a complementary angle.
Solution: cos 35° = sin 55° (since 35 + 55 = 90).
Example 5: Finding an Angle
If sin θ = cos 2θ, find θ (0 < θ < 90°).
Solution: sin θ = cos(90° − θ). So cos(90° − θ) = cos 2θ. Therefore 90° − θ = 2θ, giving 3θ = 90°, θ = 30°.
Combining Properties
Example 6: Multiple Identities
Simplify: sin(−θ) · csc θ + cos(−θ) · sec θ
Solution:
= (−sin θ)(1/sin θ) + (cos θ)(1/cos θ) = −1 + 1 = 0
Check Your Understanding
1. Simplify: tan(−π/3)
2. Express cot 40° as a trig function of a complementary angle.
3. Simplify: cos(−θ) · tan(−θ)
Key Takeaways
- Cosine and secant are even: f(−θ) = f(θ).
- Sine, tangent, cosecant, cotangent are odd: f(−θ) = −f(θ).
- Cofunctions of complementary angles are equal: sin θ = cos(90° − θ), etc.
- These properties are essential for simplifying expressions and verifying identities.