If cos0 = (3)/(√(23)) and angle 0 is in Quadrant I, what is the exact value of tan20 in simplest radical form?

Question

If cos0 =
`(3)/(√(23))` and angle 0 is in Quadrant I, what is the exact value of tan20 in simplest radical form?

Answer 1

Explanation:

If cos(θ) = 3/23 and θ is in Quadrant I, we can use the Pythagorean identity to find sin(θ):

sin^2(θ) + cos^2(θ) = 1

sin^2(θ) + (3/23)^2 = 1

sin^2(θ) = 1 - (3/23)^2

sin^2(θ) = 1 - 9/529

sin^2(θ) = 520/529

sin(θ) = √(520/529)

sin(θ) = (sqrt(520))/23

Now we can use the definition of tangent to find tan(2θ):

tan(2θ) = (2tan(θ))/(1-tan^2(θ))

We know that tan(θ) = sin(θ)/cos(θ), so:

tan(θ) = (sqrt(520))/69

tan^2(θ) = 520/4761

tan(2θ) = (2*(sqrt(520))/69)/(1 - (520/4761))

tan(2θ) = (2*(sqrt(520))/69)/(3641/4761)

tan(2θ) = (2*(sqrt(520))/69)*(4761/3641)

tan(2θ) = (9522/3641)*(sqrt(520))/69

tan(2θ) = (9522*sqrt(520))/(3641*69)

tan(2θ) = (168*sqrt(130))/529

Therefore, the exact value of tan(2θ) in simplest radical form is (168*sqrt(130))/529.