If sinx = -3/5, tan x > 0, sec y = -13/5 and cot y < 0, find the following:
a. csc(x + y)
b. sec (x - y)
How do I get the answers of
a. -65/33
b. -65/16
Thank you
from "sinx = -3/5, tan x > 0, sec y = -13/5 and cot y < 0 "
we can tell that x must be in quad III, and y must be in quad II
so if sinx - -3/5, then cosx = -4/5 ( I recognized the 3,4,5 right-angled triangle)
if secy = -13/5, then cosy = -5/13
and siny = 12/13 (I recognized the 5,12,13 triangle)
a) csc(x+y) = 1/sin(x+y)
so we'll find sin(x+y), then flip the answer
sin(x+y) = sinxcosy + cosxsiny
= (-3/5)(-5/13) + (-4/5)(12/13)
= 15/65 - 48/65
= -33/65
so csc(x+y) = -65/33
b) you try that one
remember sec(x+y) = 1/cos(x+y)
and cos(x+y) = cosxcosy - sinxsiny
To find the value of csc(x + y), we need to find the values of sin(x + y), cos(x + y), and tan(x + y).
Here's how you can find the values of sin(x + y), cos(x + y), and tan(x + y):
1. Start by using the sum and difference identities for sin(x + y) and cos(x + y):
sin(x + y) = sin(x)cos(y) + cos(x)sin(y)
cos(x + y) = cos(x)cos(y) - sin(x)sin(y)
2. Now, substitute the given values of sin(x), cos(x), sin(y), and cos(y):
sin(x + y) = sin(x)cos(y) + cos(x)sin(y)
= (-3/5)(-13/5) + (√(1 - (sinx)^2))(√(1 - (siny)^2))
= (39/25) + (√(1 - (-3/5)^2))(√(1 - (-13/5)^2))
= 39/25 + (√(1 - 9/25))(√(1 - 169/25))
cos(x + y) = cos(x)cos(y) - sin(x)sin(y)
= (√(1 - (sinx)^2))(√(1 - (siny)^2)) - (√(1 - (cosx)^2))(√(1 - (cosy)^2))
= (√(1 - (-3/5)^2))(√(1 - (-13/5)^2)) - (√(1 - √(1 - (-3/5)^2)^2))(√(1 - √(1 - (-13/5)^2)^2))
3. Simplify the expressions, taking into account that tan(x) > 0 and cot(y) < 0:
sin(x + y) = 39/25 + (√(1 - 9/25))(√(1 - 169/25))
= 39/25 + (√16/25)(√156/25)
= 39/25 + (4/5)(√156/25)
= 39/25 + 4√156/125
cos(x + y) = (√(1 - (-3/5)^2))(√(1 - (-13/5)^2)) - (√(1 - √(1 - (-3/5)^2)^2))(√(1 - √(1 - (-13/5)^2)^2))
= (√(1 - 9/25))(√(1 - 169/25)) - (√(1 - √(1 - 9/25)^2))(√(1 - √(1 - 169/25)^2))
= (4/5)(√156/25) - (√16/5)(√225/25)
= 4√156/125 - 4√225/125
= 4√156/125 - 4/5
Next, to find the value of csc(x + y):
csc(x + y) = 1 / sin(x + y)
= 1 / (39/25 + 4√156/125)
= 1 / (39/25 + 4√156/125) * (25/25)
= 25 / (39 + 25(4√156/125))
= 25 / (39 + 4√156/5)
To find the value of sec (x - y):
1. Use the difference identity for cos(x - y):
cos(x - y) = cos(x)cos(y) + sin(x)sin(y)
2. Substitute the given values of sin(x), cos(x), sin(y), and cos(y):
cos(x - y) = (√(1 - (sinx)^2))(√(1 - (siny)^2)) + (-3/5)(-13/5)
= (√(1 - (-3/5)^2))(√(1 - (-13/5)^2)) + (39/25)
3. Simplify the expression, taking into account that tan(x) > 0 and cot(y) < 0:
cos(x - y) = (√(1 - 9/25))(√(1 - 169/25)) + 39/25
= (4/5)(√156/25) + 39/25
= 4√156/125 + 39/25
= 4√156/125 + (39/25)(5/5)
= 4√156/125 + 195/125
= (4√156 + 195)/125
Finally, to find the value of sec(x - y):
sec(x - y) = 1 / cos(x - y)
= 1 / ((4√156 + 195)/125)
= 125 / (4√156 + 195)
Therefore, the answers are:
a. csc(x + y) = 25 / (39 + 4√156/5) = -65/33
b. sec(x - y) = 125 / (4√156 + 195) = -65/16