Symmetry and Periodicity of Trig Functions

In this lesson:

• Prove cosine is even and sine is odd using the unit circle
• Show that sine and cosine have period , tangent has period
• Use these properties to simplify trig expressions

What You Will Be Able to Do

By the end of this lesson, you will:

1. Explain why cosine is even:
2. Explain why sine is odd:
3. Classify tangent as odd via
4. Show sine and cosine have period
5. Show tangent has period

What Happens at Negative Angles?

You've worked with even and odd functions in algebra:

• Even: — symmetric about the -axis
• Odd: — symmetric about the origin

Question: Which category fits and ?

Think about what the unit circle shows at angle vs. angle .

Even and Odd: Symmetry Definitions

• Even: — graph symmetric about the -axis
• Odd: — graph has 180° rotational symmetry
• Most functions are neither even nor odd

Even and odd describe symmetry, not whether outputs are positive or negative.

Unit Circle: Angles and

Both points share the same -coordinate but have opposite -coordinates.

Cosine Is an Even Function

From the unit circle reflection:

• Angle → point :
• Angle → point :

Cosine is even — the -coordinate is unchanged by reflection.

Sine Is an Odd Function

From the same reflection:

• Angle → point :
• Angle → point :

Sine is odd — the -coordinate is negated by the reflection.

Verifying with Numbers:

Same cosine value confirms even. Negated sine confirms odd.

Quick Check: Even, Odd, or Neither?

Classify

Is ? Is ? Or neither?

This equals neither nor ... so is neither even nor odd.

Symmetry Done — Now What about Repetition?

Sine and cosine repeat — but how often?

After one full rotation by , the point returns to start:

Is the smallest repeat interval?

Period: The Smallest Full Cycle

The period of is the smallest positive such that:

• Starting at any angle and adding = one full rotation
• You return to the same point: same coordinates, same function values

Sine and Cosine Have Period

Is a period? Test :

is not a period of sine — it fails for . The period is .

Quick Check: What Makes a Period?

True or false?

"Since , the period of sine is ."

The conclusion is correct — but is the reasoning valid?

Think: does "sin equals zero at two points separated by " prove that is the period?

Does Tangent Also Have Period ?

Recall:

At angle :

Both coordinates flip — what happens to their ratio?

Why Tangent Repeats Every Radians

Tangent repeats every — both numerator and denominator negate, ratio unchanged.

Tangent Is Also an Odd Function

Tangent is odd — inherited from sine being odd and cosine being even.

Check: ✓

Quick Check: Use the Period of Tangent

Evaluate using the period of tangent.

Hint: subtract and see where you land.

Think about what quadrant is in before checking the answer.

Evaluating Using Period Reduction

At : reference angle , in QII where cosine is negative.

So ✓

Putting It All Together: Strategy

Steps: (1) odd/even → remove negatives; (2) period → reduce angle; (3) evaluate.

Applying Odd Property: Evaluate

Step 1: Apply odd property of sine

Step 2: is in QII, reference angle ;

Applying Period Reduction: Evaluate

Step 1: Apply period — subtract twice:

Step 2: Apply even property:

Step 3: Reference angle , QIII:

Guided Practice: Simplify Step by Step

Step 1: Sine is odd — what do you write first?

Step 2: is in QIV. Reference angle = ?

Step 3: In QIV, sine is — positive or negative?

Complete the evaluation, then advance for the full answer.

Simplify Using Odd/Even and Period Properties

Simplify each expression, labeling each property you use.

Answers: Odd/Even and Period Simplifications

1. (odd; ref , QII)

2. (period ; subtract )

3. (period ; subtract )

4. (even; input negation ignored)

Summary: Symmetry and Periodicity of Trig Functions

Even/odd is about symmetry, not sign; tangent period is , not ; period requires for all .

Coming Up Next: Modeling Periodic Phenomena

HSF.TF.B.5 — Modeling with Trig Functions

• Period becomes an adjustable parameter in
• Real phenomena (Ferris wheels, tides) need custom periods
• Today's symmetry properties simplify those models