Find sin(x+y), cos(x-y), tan(x+y), and the quadrant of (x+y) if sinx= -1/4, cosy= -4/5, with x and y in quadrant 3.
you need to sketch right-angled triangles to get the missing sides
given: sinx = -1/4, y = -1, r = 4
x^2 + y^2 = r^2
x^2 + 1 = 16
x = -√15 in III
So sinx = -1/4 , cosx = -√15/4 , tanx = 1/√15
given: cosy = -4/5, x = -4, r = 5
then y = -3
siny = -3/5, cosy = -4/5 , tany = 3/4
Now we have all the parts, let's just use the definitions of ....
sin(x+y) = sinxcosy + cosxsiny
= (-1/4)(-4/5)+ (-√15/4)(-3/5)
= (4 + 3√15)/20
cos(x-y) = cosxcosy + sinxsinx
= .....
you try this one
tan(x+y) = (tanx + tany)/(1 - tanxtany)
= (1/√15 + 3/4)/(1 - (1/√15)(3/4))
= ( (4+3√15)/(4√15) )/( (4√15 - 3)/(4√15))
= (4 + 3√15)/(4√15 - 3)
To find sin(x+y), cos(x-y), tan(x+y), and the quadrant of (x+y), we can use trigonometric identities and the given information about the values of sinx and cosy.
1. Calculate sin(x+y):
The formula to calculate sin(x+y) is:
sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
Given:
sinx = -1/4
cosy = -4/5
To find sin(x), we can use the fact that sinx = -1/4 and that x is in quadrant 3. In quadrant 3, sin is negative. Therefore, sinx is negative.
Similarly, to find cos(y), we can use the fact that cosy = -4/5 and that y is in quadrant 3. In quadrant 3, cos is negative. Therefore, cosy is negative.
Substituting these values into the formula, we get:
sin(x+y) = (-1/4)(-4/5) + (cosy)(sinx)
= 1/5 + (-1/4)
= 4/20 - 5/20
= -1/20
Therefore, sin(x+y) is equal to -1/20.
2. Calculate cos(x-y):
The formula to calculate cos(x-y) is:
cos(x-y) = cos(x)cos(y) + sin(x)sin(y)
Given:
sinx = -1/4
cosy = -4/5
Using the information about x and y being in quadrant 3, we can deduce the signs of cosx and siny. In quadrant 3, cos is negative, and sin is negative. Therefore, both cosx and siny are negative.
Substituting these values into the formula, we get:
cos(x-y) = (cosx)(cosy) + (-1/4)(-4/5)
= (-1)(-4/5) + (-1/4)(-4/5)
= 4/5 + 4/20
= 20/20 + 4/20
= 24/20
= 6/5
Therefore, cos(x-y) is equal to 6/5.
3. Calculate tan(x+y):
The formula to calculate tan(x+y) is:
tan(x+y) = (tanx + tany) / (1 - tanx*tany)
Given:
sinx = -1/4
cosy = -4/5
We already know from the earlier deductions that sinx and cosy are negative. Therefore, tanx = sinx/cosx = (-1/4)/(-√(1-(1/4)^2)) = (-1/4)/(-√(15/16)) = -1/√15.
Similarly, tany = sin(y)/cos(y) = (-√(1-(4/5)^2))/(-4/5) = (-√(1-16/25))/(-4/5) = (√(9/25))/(-4/5) = 3/(-4√5/5) = -3/(-4√5/5) = 3/(4√5/5) = 15/(4√5).
Substituting these values into the formula, we get:
tan(x+y) = (tanx + tany) / (1 - tanx*tany)
= (-1/√15 + 15/(4√5)) / (1 - (-1/√15)*(15/(4√5)))
Simplifying the expression is not possible without numerical values for calculating tan(x+y).
4. Determine the quadrant of (x+y):
Given that x and y are both in quadrant 3, the sum x+y will also be in quadrant 3.
Therefore, (x+y) is in quadrant 3.
To summarize:
sin(x+y) = -1/20
cos(x-y) = 6/5
tan(x+y) cannot be simplified without numerical values.
The quadrant of (x+y) is 3.