Eclatech Solutions | Interactive Pre-Calculus Tools Free for educational use

The Formula Rotator

Sum and difference formulas, double-angle, half-angle, and sum-to-product identities

Tune a guitar by ear and you hear a wobble -- wah-wah-wah -- that slows down as you get closer, then vanishes when the strings match. That wobble is two sine waves adding. The formula that explains it is on your identity sheet, and once you hear it you cannot unhear it.

Step 1 of 10

Is \(\sin(A + B) = \sin A + \sin B\)?

Think about it. Does the sine function distribute over addition the way multiplication does?

Step 2 of 10

Two points on the unit circle at angles A and B. What determines the chord length between them?

Use the sliders to place two points on the unit circle. Watch the chord between them.

60°
20°

The chord length depends only on the angle between the two points, which is \(A - B\). Move A and B around -- as long as the difference stays the same, the chord stays the same length.

Step 3 of 10

Compute the chord length two different ways.

Way 1: Distance formula between \((\cos A, \sin A)\) and \((\cos B, \sin B)\):

\(d^2 = (\cos A - \cos B)^2 + (\sin A - \sin B)^2\)

Way 2: Rotate so one point sits at angle 0 and the other at \(A - B\). Same chord, different coordinates:

\(d^2 = (\cos(A-B) - 1)^2 + (\sin(A-B) - 0)^2\)

The chord did not change length when we rotated. So these two expressions must be equal. Which step should we do next?

Step 4 of 10

Set them equal and simplify. Watch the algebra.

Expand Way 1:

\(\cos^2 A - 2\cos A\cos B + \cos^2 B + \sin^2 A - 2\sin A\sin B + \sin^2 B\)

Use \(\cos^2\theta + \sin^2\theta = 1\) twice:

\(= 2 - 2\cos A\cos B - 2\sin A\sin B\)

Expand Way 2:

\(\cos^2(A-B) - 2\cos(A-B) + 1 + \sin^2(A-B)\)

Use \(\cos^2\theta + \sin^2\theta = 1\):

\(= 2 - 2\cos(A-B)\)

Set equal:

\(2 - 2\cos(A-B) = 2 - 2\cos A\cos B - 2\sin A\sin B\)

Cancel the 2's and divide by \(-2\):

\(\cos(A - B) = \cos A\cos B + \sin A\sin B\)

One formula. Earned, not memorized.

Step 5 of 10

Now replace B with \(-B\). What happens?

You know from the unit circle that cosine is an even function: \(\cos(-B) = \cos B\). And sine is odd: \(\sin(-B) = -\sin B\).

Starting from \(\cos(A - B) = \cos A\cos B + \sin A\sin B\), replace every B with \(-B\):

Step 6 of 10

Use cofunctions to derive the sine formulas.

From Simulation 17, you know that \(\sin\theta = \cos(90° - \theta)\). So:

\(\sin(A + B) = \cos\bigl(90° - (A + B)\bigr) = \cos\bigl((90° - A) - B\bigr)\)

Apply the cosine difference formula with "\(90° - A\)" playing the role of the first angle:

\(= \cos(90° - A)\cos B + \sin(90° - A)\sin B\)

And the cofunctions turn those back into sine and cosine:

\(\sin(A + B) = \sin A\cos B + \cos A\sin B\)

Replace B with \(-B\) to get the difference version:

\(\sin(A - B) = \sin A\cos B - \cos A\sin B\)

Four formulas from one construction and two substitutions.

Step 7 of 10

In the sum formula, what if \(B = A\)?

Take \(\sin(A + B) = \sin A\cos B + \cos A\sin B\) and set \(B = A\). What do you get for \(\sin(2A)\)?

Step 8 of 10

The three faces of \(\cos(2A)\).

Setting \(B = A\) in \(\cos(A + B) = \cos A\cos B - \sin A\sin B\) gives:

\(\cos(2A) = \cos^2 A - \sin^2 A\)

Now use \(\sin^2 A = 1 - \cos^2 A\) to get one form, and \(\cos^2 A = 1 - \sin^2 A\) to get another.

Step 9 of 10

Half-angle from power reduction.

Take \(\cos(2A) = 1 - 2\sin^2 A\) and solve for \(\sin^2 A\):

\(\sin^2 A = \frac{1 - \cos(2A)}{2}\)

That is the power-reduction formula. Now substitute \(A = \frac{\theta}{2}\):

\(\sin^2\!\left(\frac{\theta}{2}\right) = \frac{1 - \cos\theta}{2}\)

Take the square root:

\(\sin\!\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos\theta}{2}}\)

The plus-or-minus is not decoration. The quadrant of \(\frac{\theta}{2}\) decides the sign, and it is the single most missed step in this unit.

If \(\theta = 300°\), then \(\frac{\theta}{2} = 150°\), which is in Quadrant II where sine is positive. So you would choose the positive root.

Step 10 of 10

Sum-to-product, and the guitar.

Two sine waves at 440 Hz and 444 Hz add together. Watch what happens:

The sum-to-product formula turns the sum into a product:

\(\sin\alpha + \sin\beta = 2\sin\!\left(\frac{\alpha+\beta}{2}\right)\cos\!\left(\frac{\alpha-\beta}{2}\right)\)

A fast carrier at \(\frac{440+444}{2} = 442\) Hz times a slow envelope at \(\frac{444-440}{2} = 2\) Hz. You hear 4 wobbles per second (2 Hz envelope = 4 beats, because both the crest and trough produce a loud moment).

Tune until the wobble stops. The beat frequency is the difference. 444 minus 440 is 4. You hear four wobbles a second.

The identity sheet is not a list. It is one construction with a family tree, and you can rebuild any branch from the root.

Explore

Click any formula to see how it derives from its parent. Every formula in this unit traces back to one construction: the chord on the unit circle.

Click a formula above to see its derivation.

Pick two angles and see the sum and difference formulas applied step by step.

45°
30°

Add two sine waves and see the beat pattern. The envelope pulsing is the beat frequency.

440
444

Watch \(\sin(2A)\) and \(2\sin A\cos A\) plot identically as A varies. They are the same function.

45°

Select a view above to explore the formulas interactively.

Try This

Find the exact value of \(\cos 15°\)

Which two special angles combine to make 15 degrees?

\(15° = 45° - 30°\), so we apply the cosine difference formula:

\(\cos 15° = \cos(45° - 30°) = \cos 45°\cos 30° + \sin 45°\sin 30°\)

What is the exact value?

The guitar tuning problem

Two guitar strings sound at 440 Hz and 444 Hz simultaneously.

Part 1: What frequency do you actually hear? (the average)

Part 2: How many beats per second?

Part 3: If you tighten the second string to 442 Hz, what happens to the beat rate?

Part 4: What does zero beats mean?

Verify the triple-angle identity

Show that \(\sin(3A) = 3\sin A - 4\sin^3 A\).

Hint: Write \(3A\) as \(2A + A\) and use the sum formula from Step 6 and the double-angle formulas from Step 7.

Step 1: What is \(\sin(2A + A)\) expanded using the sine sum formula?

  • Derive the cosine difference formula from the unit circle chord construction
  • Use substitution to obtain cosine sum, sine sum, and sine difference formulas
  • Apply sum and difference formulas to find exact values of non-special angles
  • Derive double-angle formulas by setting B = A in the sum formulas
  • Produce all three forms of the double-angle cosine formula
  • Derive half-angle and power-reduction formulas from the double-angle formulas
  • Apply sum-to-product formulas to explain beat frequencies

Quick Check

What is the exact value of \(\sin 75°\)?

Hint: \(75° = 45° + 30°\).

Instructor Notes

Teaching Notes

  • The chord-length construction in Steps 2-4 is the conceptual anchor. Students who internalize "rotate the chord" can rederive the cosine difference formula on an exam without memorizing it.
  • Emphasize the family tree: every identity in this unit flows from \(\cos(A - B)\). When students see the derivation path, the identity sheet stops looking like an arbitrary list.
  • The beat-frequency demonstration in Step 10 and the Wave Adder are powerful motivators. If classroom audio is available, play it live.
  • Half-angle sign errors are the most common exam mistake in this unit. Drill the question: "What quadrant is half the angle in?" separately from the formula itself.

Common Student Errors

  • Distributing sine/cosine: Treating \(\sin(A + B)\) as \(\sin A + \sin B\). Step 1 is designed to break this misconception early.
  • Sign confusion in sum vs. difference: Mixing up which formula has the plus and which has the minus. Mnemonic: cosine difference has plus (the "opposite" pattern).
  • Half-angle sign from wrong angle: Choosing the sign of \(\sin(\theta/2)\) based on the quadrant of \(\theta\) instead of the quadrant of \(\theta/2\). Step 9 addresses this directly.
  • Forgetting the plus-or-minus entirely: Writing \(\sin(\theta/2) = \sqrt{\ldots}\) without the sign. The formula produces two values and the context decides which one.

Discussion Questions

  • Why does the cosine difference formula have a plus sign while the cosine sum formula has a minus sign? Can you explain this geometrically?
  • If you only memorize \(\cos(A - B)\), how many steps does it take to get to \(\sin(A + B)\)?
  • A piano tuner hears 3 beats per second between two notes. What can you say about the frequency difference?
  • Why are there three forms of \(\cos(2A)\) but only one form of \(\sin(2A)\)?

Exam Connection

  • Finding exact values like \(\cos 15°\), \(\sin 75°\), \(\tan 105°\) using sum/difference formulas
  • Simplifying expressions using double-angle and half-angle formulas
  • Verifying identities by expanding one side into the other (the triple-angle challenge is a classic)
  • Power-reduction: converting \(\sin^4 x\) or \(\cos^2 x \sin^2 x\) into first-power cosine expressions