Given Sin(A) = ⅗ and Cos(B) = 8/17 in Quadrant I, find Sin(A+B).
To find Sin(A+B), we can use the formula for the sine of the sum of two angles:
Sin(A+B) = Sin(A) * Cos(B) + Cos(A) * Sin(B)
First, let's solve for Cos(A) using the given information. Since sin(A) = ⅗, we can use the Pythagorean identity to find Cos(A):
Cos(A) = √(1 - Sin^2(A))
= √(1 - (⅗)^2)
= √(1 - 9/25)
= √(16/25)
= 4/5
Now we have the values of Sin(A), Cos(A), and Cos(B). We can substitute these values into the formula to solve for Sin(A+B):
Sin(A+B) = Sin(A) * Cos(B) + Cos(A) * Sin(B)
= (⅗) * (8/17) + (4/5) * Sin(B)
Since we don't have the value of Sin(B), we can't calculate the exact value of Sin(A+B). However, we can still simplify the expression further.
Multiplying the fractions:
Sin(A+B) = (8/5) * (⅗ * 1/17) + (4/5) * Sin(B)
= 24/85 + (4/5) * Sin(B)
Therefore, the exact value of Sin(A+B) cannot be determined without the value of Sin(B).