Suppose that cosx= -3/5 and x is in the third quadrant, and that siny= 2/9 and y is in the second quadrant. Find the exact values

1). tan(y/2)

2). cos(x+y)

3). tan(x+y)

Draw your triangles in standard position. Then it is clear that

sinx = -4/5
tanx = 4/3

cosy = -√77/9
tany = -2/√77

Now just plug the needed values into the identities you have

To find the exact values of the given trigonometric expressions, we can use the following steps:

1). Recall that the half-angle identity for tangent is given by:
tan(x/2) = ±√((1 - cos(x)) / (1 + cos(x)))

2). To find cos(x + y), we can use the sum identity for cosine:
cos(x + y) = cos(x) cos(y) - sin(x) sin(y)

3). Similarly, for tan(x + y), we can use the sum identity for tangent:
tan(x + y) = (tan(x) + tan(y)) / (1 - tan(x) tan(y))

Now, let's find the exact values of the given expressions step by step:

1). tan(y/2):
We are given that siny = 2/9, and y is in the second quadrant. Since sine is positive in the second quadrant, we can determine that sin(y) = 2/9.
To find tan(y/2), first we need to find cos(y). Since y is in the second quadrant, we can use the Pythagorean identity to find cos(y):
cos^2(y) = 1 - sin^2(y)
cos^2(y) = 1 - (2/9)^2
cos^2(y) = 1 - 4/81
cos^2(y) = 77/81
cos(y) = ±√(77/81)

Now, to find tan(y/2), we substitute the values of cos(y) and sin(y) into the half-angle identity:
tan(y/2) = ±√((1 - cos(y)) / (1 + cos(y)))
tan(y/2) = ±√((1 - √(77/81)) / (1 + √(77/81)))

2). cos(x + y):
We are given that cosx = -3/5, and x is in the third quadrant. Since cosine is negative in the third quadrant, we can determine that cos(x) = -3/5.
To find sin(x), we can use the Pythagorean identity:
sin^2(x) = 1 - cos^2(x)
sin^2(x) = 1 - (-3/5)^2
sin^2(x) = 1 - 9/25
sin^2(x) = 16/25
sin(x) = ±√(16/25)

Now, substitute the values of sin(x) and cos(x) into the sum identity for cosine:
cos(x + y) = cos(x) cos(y) - sin(x) sin(y)
cos(x + y) = (-3/5) cos(y) - (√(16/25)) (√(77/81))

3). tan(x + y):
We can use the given values of cos(x) and sin(x) to find tan(x):
tan(x) = sin(x) / cos(x)
tan(x) = (√(16/25)) / (-3/5)
tan(x) = -√(16/25) / (3/5)
tan(x) = -√(16/25) * (5/3)
tan(x) = -4/3

Now, substitute the values of tan(x) and tan(y) into the sum identity for tangent:
tan(x + y) = (tan(x) + tan(y)) / (1 - tan(x) tan(y))
tan(x + y) = (-4/3 + 2/9) / (1 - (-4/3)(2/9))

Simplify this expression to obtain the exact value of tan(x + y).