Sum and Difference Identities

Key Questions

  • How do you apply the sum and difference formula to solve trigonometric equations?

    Main Sum and Differences Trigonometric Identities

    cos (a - b) = cos a*cos b + sin a*sin b
    cos (a + b) = cos a*cos b - sin a*sin b
    sin (a - b) = sin a*cos b - sin b*cos a
    sin (a + b) = sin a*cos b + sin b*cos a
    tan (a - b) = (tan a - tan b)/(1 + tan a*tan b)
    tan (a + b) = (tan a + tan b)/(1 -tan a*tan b)

    Application of Sum and Differences Trigonometric Identities

    Example 1: Find sin 2a.

    sin 2a
    = sin (a + a)
    = sin a*cos a + sin a*cos a
    = 2*sin a*cos a

    Example 2: Find cos 2a.

    cos 2a
    = cos (a + a)
    = cos a*cos a - sin a*sin a
    = cos^2 a - sin^2 a

    Example 3: Find cos ((13pi)/12).

    cos ((13pi)/12)
    = cos (pi/3 + (3pi)/4)
    = cos (pi/3)*cos ((3pi)/4) - sin (pi/3)*sin ((3pi)/4)
    = -(sqrt2)/4 - (sqrt6)/4
    = -[sqrt2 + sqrt6]/4

    Nghi N. · 3 · Apr 7 2015
  • What are some sum and difference identities examples?

    Here is an example of using a sum identity:

    Find sin15^@.

    If we can find (think of) two angles A and B whose sum or whose difference is 15, and whose sine and cosine we know.

    sin(A-B)=sinAcosB-cosAsinB

    We might notice that 75-60=15
    so sin15^@=sin(75^@-60^@)=sin75^@cos60^@-cos75^@sin60^@

    BUT we don't know sine and cosine of 75^@. So this won't get us the answer. (I included it because when solving problems we DO sometimes think of approaches that won't work. And that's OK.)

    45-30=15 and I do know the trig functions for 45^@ and 30^@

    sin15^@=sin(45^@-30^@)=sin45^@cos30^@-cos45^@sin30^@

    =(sqrt2/2)(sqrt3/2)-(sqrt2/2)(1/2)

    =(sqrt6 - sqrt 2)/4

    There are other way of writing the answer.

    Note 1
    We could use the same two angles and the identity for cos(A-B) to find cos 15^@

    Note 2
    Instead of 45-30=15 we could have used 60-45=15

    Note 3
    Now that we have sin 15^@ we could use 60+15=75 and sin(A+B) to find sin75^@. Although if the question had been to find sin75^@, I'd probably use 30^@ and 45^@#

    Jim H · 1 · Mar 25 2015

Questions

Trigonometric Identities and Equations