Sum and Difference Formulas for Sine and Cosine
The sum Formulas for Sine and Cosine are:
Sin(A+ B) = SinA.CosB + CosA.SinB
Cos(A+B) = CosA.CosB – SinA.SinB
The difference Formulas for Sine and Cosine are:
Sin(A – B) = SinA.CosB - CosA.SinB
Cos(A – B) = CosA.CosB + SinA.SinB
Trigonometric Sum and Difference Formulas
Sin(x+y) = sin(x).cos(y) + cos(x).sin(y)
Cos(x+y) = cos(x).cos(y) – sin(x).sin(y)
Tan(x+y) = [tan(x) + tan(y)]/[1- tan(x).tan(y)]
Sin(x-y) = sin(x).cos(y) – cos(x).sin(y)
Cos(x-y) = cos(x).cos(y) + sin(x).sin(y)
Tan(x-y) = [tan(x) – tan(y)]/[1+ tan(x).tan(y)]
Let us solve some of trigonometric problems using trig sum and difference formulas
Solve, cos(30 degrees)cos(15 degrees) – sin(30 degrees)sin(15 degrees) without actually solving. The given trigonometric expression is in the form cos(x).cos(y) – sin(x).sin(y) which is equal to cos(x+y) a trig sum formula. Comparing the terms we get, x = 30 degrees and y = 15 degrees and hence x+ y = 30 + 15 = 45 degrees. Finally we get, cos(x+y) = cos(30+15) = cos(45) = sqrt(2)/2
Sum and Difference Formulas Trig functions sine, cosine and tangent are given as follows:
Sin(alpha+ beta) = sin(alpha).cos(beta) + cos(alpha).sin(beta)
solve sin(75 degrees)
75 degrees is not special angle, but we can split 75 to give 45 + 30, we know both 45 and 30 degrees are special angles. So, we can re-write sin(75 degrees) = sin(45+30) applying the sum formula of sine, we get
Sin(45).cos(30) + cos(45).sin(30) = (1/2)(1/sqrt2) + (sqrt3/2) (1/sqrt2) = sqrt(2)[sqrt(3) +1]/4
Cos(alpha+beta) = cos(alpha).cos(beta) – sin(alpha).cos(beta)
Solve cos(5 pi/12) = cos(pi/4 + pi/6)= cos(pi/4).cos(pi/6) – sin(pi/4).sin(pi/6) = [sqrt(2)/2 ].[sqrt(3)/2] – [sqrt(2)/2. ½]= [sqrt(6) – sqrt(2)]/4
Tan(alpha+beta)
= sin(alpha+beta)/cos(alpha+beta)
= [sin(alpha)cos(beta) + cos(alpha)sin(beta)]/[cos(alpha)cos(beta) – sin(alpha).sin(beta)]
= {[sin(alpha)cos(beta)/cos(alpha)cos(beta)] +[ cos(alpha)sin(beta)/cos(alpha)cos(beta)]}
Divided by [cos(alpha)cos(beta)/ cos(alpha)cos(beta)] – [sin(alpha).sin(beta)/ cos(alpha)cos(beta)]
= [tan(alpha) + tan(beta)]/[1- tan(alpha)tan(beta)]
tan(alpha+beta) = [tan(alpha) + tan(beta)]/[1- tan(alpha)tan(beta)]
Sin(alpha- beta) = sin(alpha).cos(beta) – cos(alpha).sin(beta)
Solve sin(15) = sin(45- 30) = sin(45).cos(30) – cos(45).sin(30)
= [sqrt(2)/2].[sqrt(3)/2] – [sqrt(2)/2].[1/2] = [sqrt(6) – sqrt(2)]/4
Cos(alpha – beta) = cos(alpha).cos(beta) + sin(alpha).sin(beta)
Verify cos(alpha – pi) = - cos(alpha). Using the above difference formula for cosine we get,
Cos(alpha – pi) = cos(alpha).cos(pi) + sin(alpha).sin(pi) we know that cos(pi) = -1 and sin(pi) = 0
Substituting the values, we get, cos(alpha – pi) = - cos(alpha) + 0 = - cos(alpha) [verified]
tan(alpha – beta)
= sin(alpha – beta)/cos(alpha – beta)
= [sin(alpha)cos(beta) - cos(alpha)sin(beta)]/[cos(alpha)cos(beta) + sin(alpha).sin(beta)]
= {[sin(alpha)cos(beta)/cos(alpha)cos(beta)] -[ cos(alpha)sin(beta)/cos(alpha)cos(beta)]}
Divided by [cos(alpha)cos(beta)/ cos(alpha)cos(beta)] + [sin(alpha).sin(beta)/ cos(alpha)cos(beta)]
= [tan(alpha) - tan(beta)]/[1 +tan(alpha)tan(beta)]
tan(alpha – beta)= = [tan(alpha) - tan(beta)]/[1 +tan(alpha)tan(beta)]
