Use identities to find indicated values
tan (theta)= (12/5), pi <0<3pi/2
Find sin (2theta)
To find the value of sin(2theta) using the given information, we can use the double-angle identity for sine:
sin(2theta) = 2 * sin(theta) * cos(theta)
However, we don't have the value of cos(theta). To find it, we can use the Pythagorean identity:
sin^2(theta) + cos^2(theta) = 1
Let's start by finding the value of cos(theta) using the given information that tan(theta) = 12/5.
We know that:
tan(theta) = sin(theta) / cos(theta)
Rearranging the equation, we have:
sin(theta) = tan(theta) * cos(theta)
Substituting the given value of tan(theta) = 12/5:
sin(theta) = (12/5) * cos(theta)
Now, we can use the Pythagorean identity to solve for cos(theta):
sin^2(theta) + cos^2(theta) = 1
[(12/5) * cos(theta)]^2 + cos^2(theta) = 1
(144/25) * cos^2(theta) + cos^2(theta) = 1
[(144/25) + 1] * cos^2(theta) = 1
[144/25 + 25/25] * cos^2(theta) = 1
[169/25] * cos^2(theta) = 1
cos^2(theta) = 25/169
cos(theta) = sqrt(25/169)
cos(theta) = 5/13
Now that we have the values of sin(theta) and cos(theta), we can substitute them into the double-angle identity for sine:
sin(2theta) = 2 * sin(theta) * cos(theta)
sin(2theta) = 2 * [(12/5) * cos(theta)] * cos(theta)
sin(2theta) = 2 * [(12/5) * (5/13)] * (5/13)
sin(2theta) = 2 * (12/13) * (5/13)
sin(2theta) = 120/169
Therefore, sin(2theta) = 120/169.