find the exact value for sin 2x if cot x=4/3 when 180° < x < 270°?

Question

Could someone please show me how to answer this question?

Answer 1

Exact value for sin(2x), cot(x) = 4/3,
180 < x < 270

First off, use the double angle identity
sin(2x) = 2sin(x)cos(x)

cot(x) = 4/3, so it follows that tan(x) = 3/4.

Use SOHCAHTOA and right angle triangles to find sin(x) and cos(x).

tan(x) = 3/4 = opp/adj
opp = 3
adj = 4, so by Pythagoras,
hyp = sqrt(3^2 + 4^2) = sqrt(25) = 5

cos(x) = adj/hyp = 4/5
But cosine is negative from 180 to 270 degrees, so
cos(x) = -4/5

sin(x) = opp/hyp = 3/5
Since cotangent is positive in quadrants 1 and 3, and sine is positive in quadrants 1 and 2, it follows that the overlapping quadrant is 1.
sine is positive in quadrant 1.

sin(2x) = 2sin(x)cos(x)
= 2[3/5] [-4/5]
= [6/5] [-4/5]
= 24/25

Answer 2

180 X 2

Answer 3

If cot x = 4/3 then the we have a 3-4-5 right triangle.
Since sin2x=2sinxcosx=-2*(4/5)*(3/5)=-24/24

Answer 4

John - I just have to say HOLY CRAP - I wish I knew, what year are you in school?

I'm not senile however, I don't remember anything that difficult!

:-)

Perhaps you'll be the next person in the world who finds the cure for cancer.... Someone has too!!!!