Let ∠X and ∠Y be complementary. If sin(X) = 3/5 and cos(X) = 4/5 then what is the value of sin(Y)?
A)3/5
B) 4/5
C) 5/4
D) 5/3
Is it A?
x+y = 90
y = 90-x
siny = sin(90-x) = cosx = 4/5
the sin( of any angle) = cos(of its complement)
the fact that sinx = 3/5 was a distraction and did not have to be given.
To find the value of sin(Y), we need to use the fact that ∠X and ∠Y are complementary angles.
By definition, two angles are considered complementary if their sum is 90 degrees (π/2 radians). In this case, ∠X + ∠Y = 90° or π/2.
Given that sin(X) = 3/5, we can use the Pythagorean identity to solve for the third side of the right triangle:
cos(X) = √(1 - sin(X)^2)
=> cos(X) = √(1 - (3/5)^2)
=> cos(X) = √(1 - 9/25)
=> cos(X) = √(16/25)
=> cos(X) = 4/5
Since the cosine of X is positive, X must be in the first quadrant. Thus, Y must be in the second quadrant.
In the second quadrant, the value of sin(Y) is positive. Therefore, the correct answer is:
B) 4/5
To find the value of sin(Y), we first need to find the value of cos(Y), since sin(Y) is equal to cos(Y)'s complement.
Given that ∠X and ∠Y are complementary, we know that ∠X + ∠Y = 90°.
Since sin(X) = 3/5 and cos(X) = 4/5, we can use the Pythagorean identity:
sin^2(X) + cos^2(X) = 1
(3/5)^2 + (4/5)^2 = 9/25 + 16/25 = 25/25 = 1
Now, we can use this identity to find sin(Y):
sin(Y) = √(1 - cos^2(Y))
sin(Y) = √(1 - (4/5)^2) = √(1 - 16/25) = √(9/25) = 3/5
Therefore, the correct answer is option A) 3/5.