theta is a first quadrant angle in standard position and cos theta = 1/(sqrt10) .
Find the exact value of sin theta,
(sin theta)^2+(cos theta)^2=1;
(sin theta)^2+1/10=1;
(sin theta)^2=9/10
sin theta= 3/(sqrt10)
OR
sin theta=-3/(sqrt10)
But since theta is a first quadrant angle
==> sin theta is positive
Therefore, sin theta= 3/(sqrt10)
To find the exact value of sin(theta), we can use the identity:
sin^2(theta) + cos^2(theta) = 1
Given that cos(theta) = 1/(sqrt(10)), we can square this value to find cos^2(theta):
cos^2(theta) = (1/(sqrt(10)))^2 = 1/10
Now, we can substitute this value into the identity to solve for sin(theta):
sin^2(theta) + 1/10 = 1
To isolate sin^2(theta), we subtract 1/10 from both sides:
sin^2(theta) = 1 - 1/10 = 9/10
Taking the square root of both sides gives us:
sin(theta) = sqrt(9/10) = sqrt(9)/sqrt(10) = 3/sqrt(10)
However, to simplify the expression further, we can rationalize the denominator:
sin(theta) = (3/sqrt(10)) * (sqrt(10)/sqrt(10)) = 3sqrt(10)/10
Thus, the exact value of sin(theta) is 3sqrt(10)/10.