Given Sin(A) = ⅗ and Cos(B) = 8/17 in Quadrant I, find Cos(A+B).
Draw your triangles and see that
cosA = 4/5
sinB = 15/17
To find Cos(A+B), we can use the trigonometric identity:
Cos(A+B) = Cos(A) * Cos(B) - Sin(A) * Sin(B)
Given that Sin(A) = ⅗ and Cos(B) = 8/17, we need to find Cos(A) and Sin(B).
To find Cos(A), we can use the Pythagorean identity:
Cos^2(A) = 1 - Sin^2(A)
Substituting the value of Sin(A) into the equation:
Cos^2(A) = 1 - (⅗)^2
= 1 - 9/25
= 25/25 - 9/25
= 16/25
Taking the square root of both sides:
Cos(A) = √(16/25)
= 4/5
To find Sin(B), we can use the Pythagorean identity:
Sin^2(B) = 1 - Cos^2(B)
Substituting the value of Cos(B) into the equation:
Sin^2(B) = 1 - (8/17)^2
= 1 - 64/289
= 289/289 - 64/289
= 225/289
Taking the square root of both sides:
Sin(B) = √(225/289)
= 15/17
Now we have all the values needed to find Cos(A+B):
Cos(A+B) = (4/5) * (8/17) - (⅗) * (15/17)
= 32/85 - 45/85
= (32 - 45)/85
= -13/85
Therefore, Cos(A+B) = -13/85.