Suppose *u* is a quadrant IV angle with cos(u) = 3/5.
Suppose *v* is a quadrant IV angle with cos(v) = 12/13.
Find the exact value of sin(u-v)
I have gotten to sin120cos150 - sin150cos120 but I am not sure I am correct up to that point and I don't know where to go from there.
draw your triangles in the correct quadrantIs.
Use Pythagoras in each case.
if cosu = 3/5, in IV, then sinu = -4/5
if cosv = 12/13, then sinv = -5/13
sin(u-v)
= sinu cosv - cosu sinv
= (-4/5)(12/13) -(3/5)(-5/13)
= (-48 + 15)/65
= -33/65
437628
To find the exact value of sin(u-v), you can use the angle subtraction formula for sine:
sin(u-v) = sin(u)cos(v) - cos(u)sin(v)
First, let's find the values of sin(u), cos(u), sin(v), and cos(v) using the given information.
Since u is a quadrant IV angle with cos(u) = 3/5, we can use the Pythagorean identity to find sin(u):
sin(u) = sqrt(1 - cos^2(u))
sin(u) = sqrt(1 - (3/5)^2)
sin(u) = sqrt(1 - 9/25)
sin(u) = sqrt(25/25 - 9/25)
sin(u) = sqrt(16/25)
sin(u) = 4/5
Similarly, for v being a quadrant IV angle with cos(v) = 12/13:
sin(v) = sqrt(1 - cos^2(v))
sin(v) = sqrt(1 - (12/13)^2)
sin(v) = sqrt(1 - 144/169)
sin(v) = sqrt(169/169 - 144/169)
sin(v) = sqrt(25/169)
sin(v) = 5/13
Now, substitute these values into the angle subtraction formula:
sin(u-v) = sin(u)cos(v) - cos(u)sin(v)
sin(u-v) = (4/5)(12/13) - (3/5)(5/13)
sin(u-v) = 48/65 - 15/65
sin(u-v) = (48 - 15)/65
sin(u-v) = 33/65
So, the exact value of sin(u-v) is 33/65.
To find the exact value of sin(u-v), you can use the trigonometric identity for the difference of two angles, which states:
sin(u-v) = sin(u)cos(v) - cos(u)sin(v)
In this case, you are given the values of cos(u) and cos(v), so you need to use the Pythagorean identity to find the values of sin(u) and sin(v) first.
Since u is a quadrant IV angle, cosine is positive and sine is negative. Therefore, sin(u) can be found using the equation:
sin(u) = -√(1 - cos^2(u))
Substituting cos(u) = 3/5, we get:
sin(u) = -√(1 - (3/5)^2) = -√(1 - 9/25) = -√(16/25) = -4/5
Similarly, for angle v, since it is also in quadrant IV, sin(v) will be negative. You can use the same method to find sin(v) as follows:
sin(v) = -√(1 - cos^2(v)) = -√(1 - (12/13)^2) = -√(1 - 144/169) = -√(25/169) = -5/13
Now that you have the values for sin(u) and sin(v), you can substitute them into the formula for sin(u-v):
sin(u-v) = sin(u)cos(v) - cos(u)sin(v)
= (-4/5)(12/13) - (3/5)(-5/13)
= -48/65 + 15/65
= -33/65
Therefore, the exact value of sin(u-v) is -33/65.