Using Addition Formulas to Compute Exact Values

Lesson 2 of 2: Exact Values and Double Angle Formulas

In this lesson:

• Decompose non-standard angles into sums of known angles
• Use the addition formulas to find exact trig values
• Derive double angle and tangent addition formulas

What You Will Learn Today

By the end of this lesson, you can:

4. Use addition formulas to find exact values of non-standard angles
5. Derive double angle formulas as special cases: and
6. Derive

Hook: Can We Find sin(75°) Exactly?

is not a standard angle.

But — and we know both.

What exact values do you need to substitute?

Angle Decomposition: Building the Toolkit

Sums or differences of 30°, 45°, 60° multiples all work.

Worked Example: sin(75°) in Full

Worked Example: cos(15°) in Full

Note: — co-function identity confirmed.

Check: Find the Exact Value of tan(75°)

Use:

,

Substitute and simplify. Rationalize if needed.

Check Answer: tan(75°) = 2 + √3

Worked Example: sin(105°) = sin(60° + 45°)

Note: ✓

The Set of Computable Angles

Any angle for integers , has an exact trig value.

Examples:

In radians:

Bridge: What Happens When A = B?

The addition formula says:

Now set :

This is the double angle formula — not a new identity.

Cosine Double Angle: Three Equivalent Forms

From

Using :

Deriving the Tangent Addition Formula

Divide numerator and denominator by :

Tangent Addition: The Denominator Matters

Test: = undefined

But (defined)

They cannot be equal — the denominator is essential.

Check: Apply Double Angle Formula

, is in Quadrant I.

Find and .

First find using the Pythagorean identity.

Check Answer: sin(2A) and cos(2A)

(Quadrant I, positive)

Check: Verify cos(120°) via Double Angle

Apply with .

Compute and confirm against the known value.

Check Answer: Double Angle Matches Unit Circle

This matches from the unit circle. ✓

All Seven Addition Formulas: Reference Table