Sine and Cosine: A Hidden Symmetry

Deck 1 of 2: Foundations and Proof

In this deck:

• Why every right triangle holds a complementary pair
• How that pair links sine and cosine
• A formal proof you can reconstruct yourself

Learning Objectives for This Deck

By the end of this deck, you should be able to:

1. Define complementary angles and find them in right triangles
2. Explain why using geometry
3. Prove the complementary angle relationship step by step
4. Describe how this relationship simplifies trigonometric work

Objectives 4 and 5 — applying and computing — are in Deck 2.

What You Already Know From HSG.SRT.C.6

You can already:

• Write for any acute angle
• Write for any acute angle
• Label opposite, adjacent, and hypotenuse relative to a chosen angle

Today's question: What happens when you switch to the other acute angle?

Complementary Angles: What the Term Means

Two angles are complementary if their measures add to 90°.

• If one angle is 30°, its complement is 60°
• If one angle is 45°, its complement is 45° (itself!)
• If one angle is 70°, its complement is 20°

Every Right Triangle Has a Complementary Pair

The two acute angles in any right triangle are always complementary.

The Angle Sum Proof in Right Triangles

In any triangle, angles sum to 180°:

In a right triangle with :

The two acute angles are always complementary. Always.

Quick Check: Find the Complement

In right triangle PQR, the right angle is at R, and .

What is ?

Think about this before advancing — use the angle sum.

Answer: Angle Q Equals 55 Degrees

Check: ✓ They are complementary.

Every right triangle gives you a complementary angle pair automatically.

Now We Connect Angles to Trig Ratios

You've established: every right triangle has a complementary acute angle pair.

Now the deeper question:

If and , how do and compare?

Let's look at the same triangle from both angles' perspectives.

The Same Sides, Two Perspectives

• Side : opposite , adjacent
• Side : adjacent , opposite
• Side : hypotenuse for both angles

Trig Ratios From Angle A

With , using sides , , :

Same triangle, same sides — but computed relative to angle A.

Your Turn: Ratios From Angle B

. The sides are still , , .

But now the roles of and have swapped.

Your turn: What are and ?

Think it through using the diagram — pause before advancing.

Trig Ratios From Angle B — The Reveal

With , the roles of and swap:

Compare these to the ratios from angle A on the previous slide.

The Proof Diagram — Our Reference

From this one triangle:

• and
• and

Proof: Step 1 and Step 2

What we're proving: for any acute

Step 1: Consider right triangle with right angle at .

Step 2: Let . Since angles sum to 180° and :

Proof: Steps 3 and 4

Step 3: Label the sides: (opposite , adjacent ), (adjacent , opposite ), (hypotenuse).

Step 4: By definition of sine and cosine:

Proof: Step 5 — The Conclusion

Step 5: Since both equal :

Similarly, and , so:

This holds for any acute angle in any right triangle.

Check-In: What Made the Proof Work?

In the proof, we deduced .

Which theorem made this step valid?

a) The Pythagorean theorem
b) The triangle angle sum (angles in a triangle sum to 180°)
c) The definition of cosine
d) The complementary angle identity itself

Choose before advancing.

Answer: The Triangle Angle Sum Theorem

The triangle angle sum theorem — all interior angles of a triangle sum to 180° — is what justified .

The proof chain:

• Triangle angle sum → determines
• Definitions of sin and cos → compute ratios from both angles
• Same side plays different roles → fractions are equal → identity proved

Checking Our Proof With the 30-60-90 Triangle

Right triangle with angles 30°, 60°, 90° and sides 1, , 2:

Because 30° and 60° are complementary!

Your Turn: Verify the Second Identity

Using the same 30-60-90 triangle (sides 1, , 2):

Verify:

• What is ? (adjacent/hypotenuse from 30°)
• What is ? (opposite/hypotenuse from 60°)
• Are they equal?

Work it out before advancing.

Answer: Both Equal Root Three Over Two

From the 30-60-90 triangle with sides 1, , 2:

Both identities confirmed numerically and proved geometrically.

Watch Out: The Function Notation Trap

is sine of a single angle — not a difference of two sines.

Watch Out: Complementary Does Not Mean Equal

Complementary means the angles add to 90° — not that they're equal.

• ✓ — different angles, complementary pair
• ✗ — this is not the relationship

only when (when the complement equals itself).

Key Takeaways: Foundations and the Proof

✓ In any right triangle, the two acute angles are complementary

✓ and

✓ Proof uses: angle sum theorem + sine/cosine definitions

✓ The same side plays opposite and adjacent roles for each acute angle

Watch out:

Watch out: Complementary ≠ equal. only at

Watch out: Relationship proven for acute angles in right triangles for now

Coming Next: Deck 2 — Applications and Symmetry

In Deck 2 you will:

• Use to find trig values without a calculator
• Simplify expressions like and
• Solve equations like
• Discover the symmetry between the sine and cosine graphs on

This relationship is a tool. Deck 2 is where you use it.