The exact value of sin(v+w), given sin v = -3/5 in Quadrant III and cos w = 10/11 in Quadrant IV, is (-30 + 4√(21))/55 after applying the sine addition formula and the Pythagorean identity.
Finding the Exact Value of sin(v+w)
To find the exact value of sin(v+w), given that sin v = -3/5 and cos w = 10/11, where v is in Quadrant III and w is in Quadrant IV, we use the sine addition formula:
sin(v+w) = sin v · cos w + cos v · sin w
Since v is in Quadrant III, both sine and cosine are negative there, so cos v is also negative.
Because sin v is given as -3/5, we can use the Pythagorean identity to find cos v:
cos v = √(1 - sin² v) = √(1 - (-3/5)²) = √(1 - 9/25) = √(16/25) = -4/5
Since w is in Quadrant IV, sine is negative and cosine is positive.
Thus, to find sin w, we again use the Pythagorean identity:
sin w = -√(1 - cos² w) = -√(1 - (10/11)²) = -√(1 - 100/121) = -√(21/121) = -√(21)/11
Now we substitute the values into the sine addition formula:
sin(v+w) = (-3/5) · (10/11) + (-4/5) · (-√(21)/11) = -30/55 + 4√(21)/55 = (-30 + 4√(21))/55
Therefore the exact value of sin(v+w) is (-30 + 4√(21))/55.