If tan theta = a/b, where a and b are positive, and if theta lies in quadrant III, find sin theta
the other side is sqrt(a^2+b^2)
sine Theta= a/other side
To find the value of sin(theta), we can use the basic trigonometric identity:
sin^2(theta) + cos^2(theta) = 1.
Since we are given that tan(theta) = a/b and we know that theta lies in quadrant III, we can determine the values of sin(theta) and cos(theta) based on the signs in that quadrant.
In quadrant III, both sine and cosine are negative. However, we can use the information from the identity above to find sin(theta) using the value of cos(theta).
Let's begin by finding the value of cos(theta). We know that:
cos(theta) = 1 / sqrt(1 + tan^2(theta))
= 1 / sqrt(1 + (a/b)^2)
= 1 / sqrt(1 + a^2/b^2)
= b / sqrt(b^2 + a^2)
Now that we have the value of cos(theta), we can substitute it into the identity:
sin^2(theta) + (b / sqrt(b^2 + a^2))^2 = 1
Rearranging the equation, we have:
sin^2(theta) + b^2 / (b^2 + a^2) = 1
Multiplying through by (b^2 + a^2) to get rid of the denominator, we have:
sin^2(theta) * (b^2 + a^2) + b^2 = b^2 + a^2
Expanding the left side of the equation:
sin^2(theta) * b^2 + sin^2(theta) * a^2 + b^2 = b^2 + a^2
Since a and b are positive and theta lies in quadrant III (where sine is negative), we know that sin^2(theta) must be positive because a negative number squared becomes positive. Therefore, we can simplify the equation:
sin^2(theta) * b^2 + sin^2(theta) * a^2 + b^2 = b^2 + a^2
sin^2(theta) * (b^2 + a^2) = b^2 + a^2 - b^2
sin^2(theta) * (b^2 + a^2) = a^2
sin^2(theta) = a^2 / (b^2 + a^2)
Taking the square root of both sides:
sin(theta) = sqrt(a^2 / (b^2 + a^2))
Since a and b are positive, we can simplify further:
sin(theta) = a / sqrt(b^2 + a^2)
Therefore, the value of sin(theta) is a / sqrt(b^2 + a^2).