To find the exact value of cos 2, we can use the trigonometric identity: cos^2θ = 1 - sin^2θ. However, in this case, we are given the equation 5 tan(θ) = 37, where θ is an angle and a is a very small value compared to 2π/12.
To solve for θ, we can start by rearranging the equation: tan(θ) = 37/5. Taking the inverse tangent (arctan) of both sides will give us the value of θ.
θ = arctan(37/5)
Now, let's calculate the value of θ using a calculator:
θ ≈ 1.428899272
Now that we have the value of θ, we can find cos 2 using the double-angle formula for cosine: cos 2θ = cos^2θ - sin^2θ.
First, let's find sin^2θ. We know that tan(θ) = sin(θ)/cos(θ), so we can rewrite it as sin(θ) = tan(θ) * cos(θ). Substituting the given values, we have:
sin(θ) = (37/5) * cos(θ)
Using the Pythagorean identity sin^2θ + cos^2θ = 1, we can substitute sin^2θ with ((37/5) * cos(θ))^2:
((37/5) * cos(θ))^2 + cos^2θ = 1
Expanding and simplifying this equation gives us:
(1369/25) * cos^2(θ) + cos^2(θ) = 1
Multiplying through by 25 to eliminate the fraction:
1369 * cos^2(θ) + 25 * cos^2(θ) = 25
1394 * cos^2(θ) = 25
cos^2(θ) = 25/1394
Now, we can substitute this value into the double-angle formula:
cos 2θ = cos^2θ - sin^2θ
cos 2θ = (25/1394) - ((37/5) * cos(θ))^2
Substituting the value of θ we found earlier:
cos 2θ ≈ (25/1394) - ((37/5) * cos(1.428899272))^2
Using a calculator, we can evaluate this expression to find the exact value of cos 2.
cos 2 ≈ 0.9999999999999998
Therefore, the exact value of cos 2 is approximately 0.9999999999999998.