Special Triangles and Exact Trig Values

HSF.TF.A.3 (+)

In this lesson:

• Derive trig values at , , from special triangles
• Use unit circle symmetry to extend values to all quadrants

Learning Objectives for This Lesson

By the end of this lesson, you will:

1. Derive side ratios for 45-45-90 and 30-60-90 triangles
2. Find exact sin, cos, tan at , ,
3. Place special triangles in the unit circle
4. Use symmetry to extend values to all quadrants

Can You Find Without a Calculator?

You know that .

For a right triangle with a 45° angle:

• The two acute angles are equal (45° each)
• So the two legs must be equal

This is the 45-45-90 triangle. Let's derive its exact side ratios.

Deriving the 45-45-90 Triangle Values

Start with legs . Apply the Pythagorean theorem:

Scale to unit circle (hypotenuse ): divide every side by :

45-45-90 on the Unit Circle

Terminal point on unit circle:

Exact Trig Values at

From the unit circle point :

Quick Check: Why Is ?

Explain using the unit circle coordinates — not the formula.

Think: what does it mean geometrically for the tangent to equal 1?

Building the 30-60-90 Triangle from Scratch

Start with an equilateral triangle, side length 2. Drop an altitude:

• Bisects the base → two sides of length 1
• Bisects the top angle → two 30° angles

Result: a 30-60-90 triangle with sides 1, , 2.

30-60-90 on the Unit Circle

Same triangle, two placements: one for , one for .

Exact Values at (30°)

Scale the 30-60-90 triangle to hypotenuse = 1 (divide by 2):

Place the 30° angle at the origin → point is

Exact Values at (60°)

Same scaled triangle, 60° angle at the origin → point is

Complementary relationship:

Deriving Tangent Values from Special Triangles

Note: , not .

Your Turn: All Trig Values at

Given the 30-60-90 triangle (scaled to hypotenuse = 1), find:

• , ,

Show the derivation — start from the triangle side lengths.

Try before advancing.

Quick Check: Deriving from Scratch

Derive from the 30-60-90 triangle — show the reasoning.

The answer is — but can you show where it comes from?

Symmetry: Reflection Across the -Axis

Take a point in QI at angle . Reflect across the -axis:

This sends angle to angle (QII).

Unit Circle Symmetry: Three Reflection Patterns

Each reflection changes coordinate signs → changes trig signs.

Symmetry: Reflection Through the Origin

Reflect through the origin: both coordinates negate.

This sends angle to angle (QIII).

Symmetry: Reflection Across the -Axis

Reflect across the -axis: only negates.

This sends angle to angle (QIV).

Worked Example: Using Symmetry for

Identify: → use

Apply:

Worked Example: Using Symmetry for

Identify: → use

Apply:

Your Turn: and

For each angle, identify which symmetry relation applies:

• : write as and apply
• : write as and apply

Complete both derivations before advancing.

Quick Check: Which Relation Applies?

For : what is the sign of the result?

• (A) Same as
• (B) Opposite to
• (C) Depends on

Explain your answer using the origin-reflection argument.

Complete First-Quadrant Exact Value Reference

Complete Unit Circle with All Coordinates

Use symmetry to extend QI values to QII, QIII, QIV.

Worked: Evaluate Three Non-Standard Angles

: QIV; ref = ;

: QII; ref = ; in QII →

: QIII; ref = ; in QIII →

Key Takeaways from This Lesson

• 45-45-90: isosceles right triangle scaled to hypotenuse 1
• 30-60-90: bisect an equilateral triangle
• Three reflections extend QI values to all quadrants
• Ref angle gives magnitude; quadrant gives sign

Watch out: , not ;

Next Steps: Even/Odd and Periodicity

Coming up — HSF.TF.A.4:

• Cosine is an even function:
• Sine is an odd function:
• Both functions are periodic with period

The x-axis reflection from today — — is the first step toward the even/odd property.