Find the exact value of sin(x-y) if sinx=-3/5 in Quadrant III and cosy=5/13 in Quadrant I.
If sin(x)=-3/5 in Q3, then
cos(x)=-4/5 (using cos²(x)=1-sin²(x))
Similarly,
cos(y)=5/13 in Q1 means sin(y)=12/13
sin(x-y)=sin(x)cos(y)-cos(x)sin(y)
=-3/5*5/13 - (-4/5)*12/13
=(-15+48)/65
=33/65
Nothing
To find the exact value of sin(x-y), we will use the trigonometric identities for sin(x-y).
The trigonometric identity for sin(x-y) is:
sin(x - y) = sin(x)cos(y) - cos(x)sin(y)
First, let's find the values of cos(x) and sin(y).
Given that sin(x) = -3/5 in Quadrant III, we can use the Pythagorean identity to find cos(x):
cos(x) = sqrt(1 - sin^2(x))
= sqrt(1 - (-3/5)^2)
= sqrt(1 - 9/25)
= sqrt(16/25)
= 4/5
Given that cos(y) = 5/13 in Quadrant I, we can use the Pythagorean identity to find sin(y):
sin(y) = sqrt(1 - cos^2(y))
= sqrt(1 - (5/13)^2)
= sqrt(1 - 25/169)
= sqrt(144/169)
= 12/13
Now we can substitute these values back into the trigonometric identity for sin(x-y):
sin(x - y) = sin(x)cos(y) - cos(x)sin(y)
= (-3/5)(5/13) - (4/5)(12/13)
= -15/65 - 48/65
= -63/65
Therefore, the exact value of sin(x-y) is -63/65.
To find the exact value of sin(x-y), we need to use the formulas for the sine of the difference of two angles.
The formula for sin(x-y) is sin(x)cos(y) - cos(x)sin(y).
First, we are given that sin(x) = -3/5 in Quadrant III. Since sin(x) is negative in Quadrant III, we can determine that cos(x) will be negative as well. To find the exact value of cos(x), we can use the Pythagorean identity: sin^2(x) + cos^2(x) = 1.
Since we know that sin(x) = -3/5, we can substitute this value into the equation:
(-3/5)^2 + cos^2(x) = 1
Simplifying,
9/25 + cos^2(x) = 1
cos^2(x) = 1 - 9/25
cos^2(x) = 16/25
cos(x) = ±√(16/25)
cos(x) = ±4/5
Since we are in Quadrant III, cos(x) must be negative, so cos(x) = -4/5.
Now, we are also given that cos(y) = 5/13 in Quadrant I. We can use the same method to find sin(y):
sin^2(y) + cos^2(y) = 1
sin^2(y) = 1 - 25/169
sin^2(y) = 144/169
sin(y) = ±√(144/169)
sin(y) = ±12/13
Since we are in Quadrant I, sin(y) must be positive, so sin(y) = 12/13.
Now, we can substitute these values into the formula sin(x-y) = sin(x)cos(y) - cos(x)sin(y).
sin(x-y) = (-3/5)(5/13) - (-4/5)(12/13)
= (-3/5)(5/13) + (4/5)(12/13)
= -15/65 + 48/65
= 33/65
Therefore, the exact value of sin(x-y) is 33/65.