Use identities to find indicated values
tan (theta)= (12/5), pi <0<3pi/2
Find sin (2theta)
To find the value of sin(2θ), we can use the double angle formula for sin:
sin(2θ) = 2sin(θ)cos(θ)
First, we need to find the values of sin(θ) and cos(θ). Given that tan(θ) = 12/5, we can use the identity:
tan^2(θ) + 1 = sec^2(θ)
Rearranging the equation, we have:
sec^2(θ) - tan^2(θ) = 1
Substituting the value of tan(θ) = 12/5, we can solve for sec(θ):
sec^2(θ) - (12/5)^2 = 1
sec^2(θ) - 144/25 = 1
sec^2(θ) = 169/25
sec(θ) = √(169/25)
sec(θ) = 13/5
Since sec(θ) = 1/cos(θ), we can find cos(θ):
1/cos(θ) = 13/5
cos(θ) = 5/13
Now that we have the values of sin(θ) = tan(θ)/sec(θ) and cos(θ), we can calculate sin(2θ):
sin(2θ) = 2sin(θ)cos(θ)
= 2 * (12/5) * (5/13)
= 24/13
Therefore, sin(2θ) = 24/13.