Consider this equation.
tan(0)= -(5)/(3)
If 0 is an angle in quadrant II, what is the value of sin(0)?
A. (3\sqrt(34))/(34)
B. (5\sqrt(34))/(34)
C. -(5\sqrt(34))/(34)
D. -(3\sqrt(34))/(34)
Since 0 is an angle in quadrant II, both sine and cosine will be negative. Specifically, cosine will be negative because it is the x-coordinate in quadrant II, and sine will be negative because it is the y-coordinate in quadrant II.
From the given equation tan(0) = -(5/3), we can use the definition of tangent:
tan(0) = sin(0) / cos(0)
In quadrant II, sin(0) will be negative and cos(0) will be negative, so we can write:
-(5/3) = sin(0) / -(cos(0))
From this, we can solve for sin(0):
sin(0) = -(5/3) * -(cos(0)) = (5/3) * cos(0)
Now, we need to find the value of cos(0) in order to determine sin(0). Since cosine is the x-coordinate in quadrant II, we can use the Pythagorean identity:
sin^2(0) + cos^2(0) = 1
Since sin(0) and cos(0) are both negative:
sin^2(0) = -(-1 - cos^2(0))
sin^2(0) = 1 - cos^2(0)
Substituting (5/3 cos(0)) for sin(0):
(25/9 cos^2(0)) = 1 - cos^2(0)
25 cos^2(0) = 9 - 9 cos^2(0)
34 cos^2(0) = 9
cos^2(0) = 9/34
cos(0) = sqrt(9/34) = 3sqrt(34) / 34
Now, substitute the value of cos(0) back into the equation for sin(0):
sin(0) = (5/3) * cos(0) = (5/3) * (3sqrt(34) / 34) = 5sqrt(34) / 34
Therefore, the value of sin(0) is (5sqrt(34))/34.
The correct answer is B. (5sqrt(34))/34.