Final answer:
To evaluate tan(2theta) given cos(theta) = 8/17 in quadrant 1, we use the Pythagorean identity to find sin(theta) and then apply the double angle formula for tangent to find tan(2theta) = -15/32.
Step-by-step explanation:
To evaluate the expression tan(2theta) under the given conditions where cos(theta) = 8/17 and theta is in quadrant 1, we need to use trigonometric identities.
First, we can determine the value of sin(theta) in the first quadrant using the Pythagorean identity:
- sin^2(theta) = 1 - cos^2(theta) = 1 - (8/17)^2
Since theta is in the first quadrant, where both sine and cosine are positive, we take the positive square root:
- sin(theta) = sqrt(1 - (64/289)) = sqrt(225/289) = 15/17
Now, we'll use the double angle formula for tangent:
- tan(2theta) = 2 * tan(theta) / (1 - tan^2(theta))
And since tan(theta) = sin(theta)/cos(theta), we can substitute:
- tan(theta) = (15/17) / (8/17) = 15/8
Finally, substituting into the double angle formula, we get:
tan(2theta) = 2 * (15/8) / (1 - (15/8)^2)
= 30/8 / (1 - 225/64)
= 30/8 / (64/64 - 225/64)
= 30/8 / (-161/64)
= -30/8 * 64/161 = -480/1288 = -15/32
So, tan(2theta) = -15/32 when cos(theta) = 8/17.