Lecture Number 12 February 16, 2009 Section 7.4 Cosine: Cos(A + B) = cosAcosB ? sinAsinB Cos(A ? B) = cosAcosB ? sinAsinB The second formula implies the first: Cos(A + B) = cos(A ? ( - B) ) = cosAcos(-B) + sinAsin(-B) cos(-B) = cos(B) and sin(-B) = -sin(B) = cosAcosB - sinAsinB Sine: Sin(A + B) = sinAcosB + cosAsinB Sin(A ? B) = sinAcosB ? cosAsinB Proof: Sin(A + B) = cos (?/2 ? (A + B)) = (?/2 ? A) ? B = cos( ( ?/2 ? A ) ? B ) Tangent: Tan(A + B) = Tan(A ? B) = Example) Use one of the equations above to solve sin(?/12) The first step is to change ?/12 into one of the equations we?ve previously learned such as ?, ?/2, ?/4, ?/3, ?/6, or 2 ?. Well, looking at ?/12, it is the same as (?/3 ? ?/4). Therefore, the equation would be the same as solving sin(?/3 ? ?/4). After we do this, we need to transform it into one of the equations above. Well, ?/3 would be A and ?/4 would be B and it would fit into the equation Sin(A ? B) = sinAcosB ? cosAsinB. Therefore, it would transform to sin(?/3)cos(?/4) ? cos(?/3)sin(?/4). Now, we already know the sine and cosine to ?/3 and ?/4 so we would change the equation to the actual values and it would look like: . Now, we would just solve by multiplying and subtracting and the answer would end up being: