## THE RHYTHM OF TRIG SUM & DIFFERENCE IDENTITIES

**Sin (A + B) = Sin A Cos B + Cos A Sin B**

**Sin (A - B) = Sin A Cos B - Cos A Sin B**

**Cos (A + B) = Cos A Cos B - Sin A Sin B**

**Cos (A - B) = Cos A Cos B + Sin A Sin B**

**Tan (A + B) =**

__Tan A + Tan B__**1 - Tan A Tan B**

**Tan (A - B) =**

__Tan A - Tan B__**1 + Tan A Tan B**

Play the notes and try to identify the song:

Did you recognize

__Mary Had a Little Lamb__? Even without the quarter notes, half notes, and measures, you would still know the tune.

Try this one with just the pacing:

__Row-Row-Row Your Boat__. If you recognized it, try using rhythm to remember the Trig Sum and Difference Identities.

First, establish the pattern of syllables. Then clap or tap the syllables for each word.

Sin (A +/- B) = Sin A Cos B +/- Cos A Sin B

Read this as SINE COSINE COSINE SIN

Tap it out as CLAP CLAP-CLAP CLAP-CLAP CLAP

Once you know the rhythm you can write out sin and cos, fill in the A’s, B’s, and plus or minus symbols.

Cos (A +/- B) = Cos A Cos B -/+ Sin A Sin B

Read as COSINE COSINE SINE SINE

Tap it out as CLAP-CLAP CLAP-CLAP CLAP CLAP

and fill in the angles and minus or plus.

Notice with these two identities, Sine sum or difference start with ‘sine’ and Cosine sum or difference start with ‘cosine’. In addition, the Sine formulas have the same sign (+ or -) as the computation of the angles, while Cosines are opposite.

Tan (A +/- B) =

__Tan A +/- Tan B__

1 -/+ Tan A Tan B

Read as TAN TAN ONE TANTAN

Tap it out as CLAP CLAP CLAP-CLAP-CLAP

When filling in operations, notice that Tangent is Sine over Cosine, so the numerator maintains the operation on the angles but the denominator uses the opposites.

Once you know these identities, you won't have to learn the Double Angle Identities because you could just substitute (A + A) for (A + B). Each Sum and Difference equation can do double duty. And who wouldn't like the requirement to learn half as much?!?

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**PRACTICE**

These exercises obviously take liberties with musical notations, but tap out the rhythms and use an unexpectedl part of your brain to remember the Sum and Difference Identities.TANGENT:

Tan Tan OneTanTan

SINE:

Sine Cosine Cosine Sine

COSINE: Cosine Cosine Sine Sine

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