To find sin(x+y), we can use the trigonometric identity:
sin(x+y) = sin(x)β
cos(y) + cos(x)β
sin(y)
Given:
sin(x) = -3/5 (x is in the 3rd quadrant, meaning sin(x) is negative)
cos(y) = -7/25
sin(y) = ?
Since sin(x) is negative and x is in the 3rd quadrant, we can use the Pythagorean theorem to find cos(x):
cos(x) = sqrt(1 - sin^2(x))
cos(x) = sqrt(1 - (-3/5)^2)
cos(x) = sqrt(1 - 9/25)
cos(x) = sqrt(16/25)
cos(x) = 4/5 (positive in the 3rd quadrant)
Now, we can find sin(y) using the Pythagorean identity:
sin^2(y) = 1 - cos^2(y)
sin^2(y) = 1 - (-7/25)^2
sin^2(y) = 1 - 49/625
sin^2(y) = 576/625
sin(y) = sqrt(576/625)
sin(y) = 24/25 (positive in any quadrant)
Substituting the values back into the trigonometric identity:
sin(x+y) = (-3/5)β
(-7/25) + (4/5)β
(24/25)
sin(x+y) = 21/125 + 96/125
sin(x+y) = 117/125
To find cos(x-y), we can use a similar approach:
cos(x-y) = cos(x)β
cos(y) + sin(x)β
sin(y)
Substituting the known values:
cos(x-y) = (4/5)β
(-7/25) + (-3/5)β
(24/25)
cos(x-y) = -28/125 - 72/125
cos(x-y) = -100/125
cos(x-y) = -4/5
To find tan(x+y), we can use the identity:
tan(x+y) = sin(x+y) / cos(x+y)
Substituting the value of sin(x+y) and cos(x-y):
tan(x+y) = (117/125) / (-4/5)
tan(x+y) = (117/125) * (-5/4)
tan(x+y) = -585/500
tan(x+y) = -117/100
To determine the quadrant of x+y, we need the signs of sin(x+y) and cos(x-y). Since sin(x+y) is positive (117/125) and cos(x-y) is negative (-4/5), the value of x+y is in the 2nd quadrant.