If Cos(θ)=12/13 it means that we can form a triangle with hypotenuse 13 and adjacent length 12. Using them to find the opposite length with the pythagorean theorem, we have:
[tex]\begin{gathered} a^2+b^2=c^2 \\ (12)^2+b^2=13^2\text{ (Replacing the values)} \\ 144+b^2=169\text{ (Raising the numbers to the power of 2)} \\ b^2=25\text{ }(\text{Subtracting 144 from both sides of the equation)} \\ b=5\text{ (Taking the square root of both sides)} \\ \text{The opposite length is 5} \end{gathered}[/tex]
If the opposite length is 5 and the angle is in the fourth quadrant then sin(θ)=-5/13.
Given that sin(2θ)=2sin(θ)cos(θ), then sin(2θ)=2(-5/13)(12/13)= -120/169.
Given that cos(2θ)=cos^2(θ)-sin^2(θ), then cos(2θ)=(12/13)^2 - (-5/13)^2 =119/169
Answers:
cos(θ)=12/13
sin(θ)=-5/13
sin(2θ)=-120/169
cos(2θ)=119/169
