Question

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

Answer 1

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`