if x and y are acute angles such that csc x = 5/4 and cos y = 15/17, find the exact value of sin (x-y), cos(x-y), cos 2y, sin 2x

Answer 1

Using SOHCAHTOA, since cscx = 5/4, sinx = 4/5 and cosx = 3/5

Since cosy = 15/17, siny = 8/17

So, sin(x-y) = sinxcosy - cosxsiny = (4/5)(15/17) - (3/5)(8/17) = 60/85 - 24/85 = 36/85

Use the following identities to do the rest:

cos(x-y) = cosxcosy + sinxsiny

cos(2y) = cos²y - sin²y

sin(2x) = 2sinxcosx

Answer 2

cscx=5/4=1.25=1/sin1.25

x=45.84...

cosy=15/17=.882...

y=28.072...

sin(x-y)=.305...

cos " =.952...

cos2y=.557...

sin2x=.999...

Answer 3

sin(x-y) = sinx cosy - cosx siny

cos(x-y) = cosx cosy + sinx siny

cos2y = 2(cosy)^2 - 1

sin2x = 2sinxcosx

So we just need to find sinx, cosx, siny and cosy and plug the values into the above formulas:

sinx = 1/cscx, so if csc x = 5/4, then sinx = 4/5

sin(x)^2 + cos(x)^2 = 1, so 16/25 + cos(x)^2 = 1, and cos(x)^2 = 9/25 and cosx = 3/5

We are given cosy = 15/17.

Similarly sin(y)^2 + cos(y)^2 = 1, so sin(y)^2 + (15/17)^2 = 1, and sin(y)^2 = 64/289 and siny = 8/17