If cos u = 4/5 and tan v = - 12/5 while (3pi)/2 < u < 2pi and pi/2 < v < pi , find : sin(u-v)

Expand sin(u - v) using the angle sum formula for sine:

sin(u - v) = sin(u) cos(v) - cos(u) sin(v)

Recall the Pythagorean identity: for all x,

cos²(x) + sin²(x) = 1

Dividing both sides by cos²(x) gives

1 + tan²(x) = sec²(x)

(since tan(x) = sin(x)/cos(x) and sec(x) = 1/cos(x))

With 3π/2 < u < 2π, we expect sin(u) < 0, and with π/2 < v < π, we expect cos(v) < 0 and sin(v) > 0.

Solve for sin(u) :

sin²(u) = 1 - (4/5)²

sin(u) = - √(1 - 16/25)

sin(u) = - √(9/25)

sin(u) = -3/5

Solve for sec(v) :

sec²(v) = 1 + (-12/5)²

sec(v) = - √(169/25)

sec(v) = -13/5

Take the reciprocal of both sides to solve for cos(v) :

1/sec(v) = 1 / (-13/5)

cos(v) = -5/13

Solve for sin(v) :

sin²(v) = 1 - cos²(v)

sin(v) = √(1 - (-5/13)²)

sin(v) = √(144/169)

sin(v) = 12/13

Now put everything together:

sin(u - v) = (-3/5) (-5/13) - (4/5) (12/13) = -33/65

geometry: triangles

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call the larger triangle abc

then the smaller triangle cde is similar to abc (all angles are equal, aaa).  

ab is the side that is 36, and we wish to find de, the side corresponding to ab

area_cde = area_abc - area_cde --> area_abc / area_cde = 2

let x=length of de

ratio of areas = (ratio of sides)^2

+2+=+%2836%2fx%29%5e2+

+2+=+1296%2fx%5e2+

++x%5e2+=+1296%2f2+=+648+

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