Given cot x= 3/4 and cosec y= -13/5 where x and y are in the same quadrant . Without finding the angles x and y, determine the exact values of sin(2x)- cos y/sec x.

Answer 1

exact answer = 132/325 = approximately 0.41

It might help to graph a unit circle and sketch the angles.

cotx = 3/4

tanx = 4/3 It's a 3-4-5 right triangle

x is in the 3rd quadrant where sinx<0

sinx = -4/5, cosx = -3/5, secx =-5/3

cscy = -13/5 It's a 5-12-13 right triangle.

y is in the 3rd quadrant. where cosy <0

siny = -5/13, cosy = -12/13

sin2x = 2sinxcosx = 2(-4/5)(-3/5) = 24/25

cosy/secx = (-12/13)/(-5/3) = (-12/13)(-3/5) = 36/65

sin2x - cosy/secx = 24/25 - 36/65 = 312/325 - 180/325 = 132/325 = 0.41

check the answer by calculating x and y

x= about 53.13+ 180 = 233.13, 2x = -.466.26 sin466.26 = sin106.23 = .96

secx = 1/cosx = 1/cos233.13 = 1/-..6 = -5/3

y = 22.62 + 180 = 202.62

cos202.62 = -.92

sin466.26 - cos202.62/sec233.13 = .96 - (-.92)/(-5/3) = .96 - .55 = 041

exact answer is 132/325

when you work a problem two different ways and get the same answer, you're virtually certain the answer is correct. 132/325 is correct to a nearly 100% probability. it's more than "just" a "check" on the answer.

Answer 2

We in quadrant III. siny = - 5/13, cosy = - √1 - (- 5/13)² = - 12/13.

cosecx = - √1+ 9/16 = - 5/4; sinx = - 4/5, cosx = - 3/5

sin(2x) - cosy/secx = 2sinxcosx - cosycosx. = -3/5(- 85 + 12/13) = 132/325

Answer 3

The terminal point of x is (-3,-4) and the radius (hypotenuse) is 5.

The terminal point of y is -5,-12) and the radius (hypotenuse) is 13.

The sin of x is-4/5 and the cosine of x is -3/5.

The sin of y is -5/13 and the cos of y is -12/13.

From these values you can calculate the value of the required expression.