-Find sin(x+y), cos(x-y), tan(x+y) and the quadrant of x+y

sinx=-3/5, cosy=-7/25, x and y in Q3

To find sin(x+y), cos(x-y), tan(x+y), and the quadrant of x+y using the given information:

Step 1: Recall the trigonometric identities:
- sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
- cos(x-y) = cos(x)cos(y) + sin(x)sin(y)
- tan(x+y) = [tan(x) + tan(y)] / [1 - tan(x)tan(y)]

Step 2: Find sin(y) and cos(y) using the given information. Since y is in Quadrant 3, sin(y) and cos(y) will be negative.
- sin(y) = -√(1 - cos^2(y)) = -√(1 - (-7/25)^2) = -24/25
- cos(y) = -7/25

Step 3: Find sin(x) and cos(x) using the given information. Since x is also in Quadrant 3, sin(x) will be negative, and cos(x) will be positive.
- sin(x) = -3/5
- cos(x) = √(1 - sin^2(x)) = √(1 - (-3/5)^2) = 4/5

Step 4: Calculate sin(x+y):
- sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
= (-3/5)(-7/25) + (4/5)(-24/25)
= 21/125 - 96/125
= -75/125
= -3/5

Step 5: Calculate cos(x-y):
- cos(x-y) = cos(x)cos(y) + sin(x)sin(y)
= (4/5)(-7/25) + (-3/5)(-24/25)
= -28/125 + 72/125
= 44/125

Step 6: Calculate tan(x+y):
- tan(x+y) = [tan(x) + tan(y)] / [1 - tan(x)tan(y)]
= [(-3/5) + (-24/25)] / [1 - (-3/5)(-24/25)]
= (-15/25 - 24/25) / (1 + 72/125)
= -39/25 / 197/125
= -39/25 * 125/197
= -5075/4925
= -1015/985

Step 7: Determine the quadrant of x+y:
Since both x and y are in Quadrant 3, the angle x+y will also be in Quadrant 3.

To find sin(x+y), we can use the trigonometric identity:

sin(x+y) = sin(x)⋅cos(y) + cos(x)⋅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)⋅(-7/25) + (4/5)⋅(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)⋅cos(y) + sin(x)⋅sin(y)

Substituting the known values:

cos(x-y) = (4/5)⋅(-7/25) + (-3/5)⋅(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.

In III , if sinx = -3/5, cosx = -4/5 , tanx = 3/4

in III, if cosy = -7/25, siny = -24/25 , tany = 24/7

( You should recognize the right-angled triangles with sides
3-4-5 and 7-24-25 )

sin(x+y)
= sinxcosy + cosxsiny
= (-3/5)(-7/25) + (-4/5)(-24/25)
= (21 + 96)/125
= 117/125

cos(x-y) = cosxcosy + sinxsiny
= (-4/5)(-7/25) + (-3/5)(-24/25)
= 28/125 + 72/125
= 100/125
= 4/5

tan(x+y) = (tanx + tany)/(1 - tanxtany)
= (3/4 + 24/7)/(1 - (3/4)(24/7))
= (117/28) / (1 - 18/7)
= (117/28) / (-11/7)
= (117/28)(-7/11)
= -117/44

sin(x+y) is positive and tan(x+y) is negative
the only quadrant where the sine is (+) and the tangent is (-) is quadrant II

x+y is in the second quadrant.