If cos A 12 and sin B 5 squareroot 2626 with angles A and

If cos A = 1/2 and sin B = -5 squareroot 26/26, with angles A and B in Quadrant IV, find the exact value of cos(A + B). Find the exact value of cos(A - B)if tan A = 1/2 and sin = 3/5, where -90 degree

Solution

cosA=1/2 and sinB = -5sqrt(26)/26 given A, and B are in IV Quadrant

cos(A+B)=cosA.cosB - sinA.sinB

to do this first we don\'t have cosB ,sinA values so we have to find them first

sin^2 A+cos^2 A=1

sin^2 A+(1/2)^2=1

sin^2 A= 1-1/4

sin^2 A=3/4

SinA = -sqrt(3)/2 [ Negative sign ,since A is in IV quadrant]

sin^2 B +cos^2 B=1

[ -5 sqrt(26)/26 ]^2 +cos^2 B=1

25 *26 /26*26 +cos^2 B=1

25/26 +cos^2 B=1

cos^2 B=1-25/26

cos^2 B=1/26

cosB = 1/sqrt(26)

now we got all values so plug in the formula

cos(A+B)=cosA.cosB - sinA.sinB

= 1/2 . 1/sqrt(26) - (-sqrt(3)/2) . -5sqrt(26)/26)

=1/2sqrt(26) - 5sqrt(3)/2sqrt(26)

= (1- 5sqrt(3)/2sqrt(26)

2). cos(A-B) if tanA= 1/2 and sinB =3/5

A, and B are in I quadrant

sec^2 A -tan^2 A=1

sec^2 -(1/2)^2 =1

sec^2 A= 1+1/4

sec^2 A= 5/4

secA = sqrt(5)/2

cosA= 2/sqrt(5)

then SinA = 1/sqrt(5)

sin^2 B +cos^2 B=1

(3/5)^2 +cos^2 B=1

9/25 +cos^2 B=1

cos^2 B =1-9/25

cos^2 B =16/25

cosB = 4/5

cos(A-B) =cosA.cosB +sinA.sinB

= 2/sqrt(5) . 4/5 +1/sqrt(5) .3/5

= 8/5sqrt(5) +3/5sqrt5)

= 11/5sqrt(5)