Question

If sin(x) = 1/4, then what is tan(2x)?
a) 1/7
b) 1/15
c) 1/3
d) 1/8

Answer 1

The tan(2x) when sin(x) = 1/4 is found using trigonometric identities, but the result does not match any of the provided options, implying a potential error in the options or the calculation.

Step-by-step explanation:

If we know that sin(x) = 1/4, we can find tan(2x) using a combination of trigonometric identities. First, we can use the double angle formula for tangent:

tan(2x) = 2tan(x) / (1 - tan^2(x))

Since sin(x) = 1/4, we need to find cos(x) and then tan(x). Using the Pythagorean identity, sin^2(x) + cos^2(x) = 1, we can find cos(x):

cos(x) = sqrt(1 - sin^2(x)) = sqrt(1 - (1/4)^2) = sqrt(1 - 1/16) = sqrt(15/16) = sqrt(15)/4

Now, tan(x) = sin(x) / cos(x) = (1/4) / (sqrt(15)/4) = 1/sqrt(15). Simplify further to rationalize the denominator:

tan(x) = sqrt(15)/15

Plugging this into the double angle formula for tangent:

tan(2x) = 2(sqrt(15)/15) / (1 - (sqrt(15)/15)^2)

tan(2x) = (2sqrt(15)/15) / (1 - 15/225)

tan(2x) = (2sqrt(15)/15) / (210/225)

tan(2x) = (2sqrt(15)/15) * (225/210)

tan(2x) = sqrt(15)/7

None of the options given match this result, so either there is an error in the options provided or in the calculation.