TRIG SUM AND DIFFERENCE IDENTITIES

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

It’s no surprise to find a teenager listening to music while studying.  There are two ways to recognize a familiar tune: the notes and the rhythm.  Try it. 

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:

 The "merrily, merrily, merrily, merrily" probably gave away that this one is 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