How do you apply the sum and difference formula to solve trigonometric equations?
1 Answer
Main Sum and Differences Trigonometric Identities
cos (a - b) = cos a*cos b + sin a*sin bcos(a−b)=cosa⋅cosb+sina⋅sinb
cos (a + b) = cos a*cos b - sin a*sin bcos(a+b)=cosa⋅cosb−sina⋅sinb
sin (a - b) = sin a*cos b - sin b*cos asin(a−b)=sina⋅cosb−sinb⋅cosa
sin (a + b) = sin a*cos b + sin b*cos asin(a+b)=sina⋅cosb+sinb⋅cosa
tan (a - b) = (tan a - tan b)/(1 + tan a*tan b)tan(a−b)=tana−tanb1+tana⋅tanb
tan (a + b) = (tan a + tan b)/(1 -tan a*tan b)tan(a+b)=tana+tanb1−tana⋅tanb
Application of Sum and Differences Trigonometric Identities
Example 1: Find
sin 2asin2a
= sin (a + a)=sin(a+a)
= sin a*cos a + sin a*cos a=sina⋅cosa+sina⋅cosa
= 2*sin a*cos a=2⋅sina⋅cosa
Example 2: Find
cos 2acos2a
= cos (a + a)=cos(a+a)
= cos a*cos a - sin a*sin a=cosa⋅cosa−sina⋅sina
= cos^2 a - sin^2 a=cos2a−sin2a
Example 3: Find
cos ((13pi)/12)cos(13π12)
= cos (pi/3 + (3pi)/4)=cos(π3+3π4)
= cos (pi/3)*cos ((3pi)/4) - sin (pi/3)*sin ((3pi)/4)=cos(π3)⋅cos(3π4)−sin(π3)⋅sin(3π4)
= -(sqrt2)/4 - (sqrt6)/4=−√24−√64
= -[sqrt2 + sqrt6]/4=−√2+√64
