find exact value if 0<x<pi/2 and 0<y<pi/2 sin(x+y) if cosx=8/17 and siny=12/37?

Answer 1

cos(x) = 8/17 = adj/hyp
adj = 8
hyp = 17, so by Pythagoras,
opp = sqrt(17^2 - 8^2) = sqrt(225) = 15

sin(x) = opp/hyp = 15/17; positive because it is in the first quadrant.

sin(y) = 12/37 = opp/hyp
opp = 12
hyp = 37, so
adj = sqrt(37^2 - 12^2) = sqrt(1225) = 35
cos(y) = adj/hyp = 35/37

sin(x + y) = sin(x)cos(y) + sin(y)cos(x)
= (15/17)(35/37) + (12/37)(8/17)
= (15*35 + 12*8) / [ 17*37 ]
= (525 + 84) / (629)
= 609/629

Answer 2

merely wreck up the fundamental at element t. F(x) = a million/[4 sin x - 3 cos x] t ? F(x) dx + -?/2 ?/2 ? F(x) dx = t (After comparing the bounds from the left and precise) (-2/5) ln(2) - ?i/5 The imaginary section comes from the indefinite fundamental being: (a million/5) * (ln[ (3 * tan(x/2) - a million) / (tan(x/2) + 3)]) overlook with regards to the scaling element for the 2d and look at each term: @ x = ?/2 ? -ln(2) @ x = -?/2 ? ln(2) + i? Now: -ln(2) - (ln(2) + i? ) = -2ln(2) - ?i there is purely one term that has the ln of a unfavourable variety. hi Dr. D. actually Maple does not like the fundamental any further than Mathematica (curiously) does. i'm integrating from -?/2 to arctan(3/4), then including to that the fundamental from arctan(3/4) to ?/2 by potential of utilising a dummy variable then taking the decrease. this is barely the 2d era that provides the imaginary section. It in order that happens that Maple will discover the indefinite fundamental, which while evaluated on the endpoints provides an analogous consequences because of fact the previous technique. I won't have greater time to place up till later, yet while the question continues to be open i will tutor the outcomes for the intermediate calculations. Later.... O.ok. So I went back and regarded returned at what I had carried out. Dr. D is right, I ignored an indication. the applicable answer is as he reported: (-2/5) ln(2) thank you Dr. D for assisting me discover that blunders! Later nonetheless.... I discovered the thank you to stress Maple to evaluate this fundamental, i ponder whether Mathematica has an identical determination. attempt this (in case you have Maple) int(a million/(4*sin(x)-3*cos(x)), x = -(a million/2)*Pi .. (a million/2)*Pi, 'CauchyPrincipalValue') devoid of the command to discover the significant fee, Maple says the fundamental is undefined. for this reason I on the beginning up grew to become into doing it by potential of hand, which bring about my blunders above. i'm optimistic chuffed for this discovery, this is going to be relatively smart sooner or later. i'd guess Mathematica has an identical determination. thank you.

Answer 3

EXACT value?

sin(x+y) = sin(x)cos(y) + cos(x)sin(y) {SINE RULE}

= SQRT(1-cos(x)^2).SQRT(1-sin(y)^2)) + (8/17).(12/37)

= SQRT(1-64/289).SQRT(1-144/1369) + 96/629

= SQRT(225/289).SQRT(1225/1369) + 96/629

= (15/17).(35/37) + 96/629

= 525/629 + 96/629

= 621/629