What is the exact value of Tan(pi/12)

We have

0 < π/12 < π/2

and

tan(π/12) = sin(π/12) / cos(π/12)

Both sin(x) and cos(x) are positive for 0 < x < π/2, so we expect tan(π/12) to also be positive.

Recall the double-angle identities for cosine,

cos²(x) = (1 + cos(2x)) / 2

as well as the Pythagorean identity,

tan²(x) = sec²(x) - 1

and by definition,

sec(x) = 1 / cos(x)

Putting everything together, we have

tan(x) = √(1 / cos²(x) - 1)

tan(x) = √(2 / (1 + cos(2x)) - 1)

Let x = π/12. Then 2x = π/6, and cos(π/6) = √3 / 2, so that

tan(π/12) = √(2 / (1 + cos(π/6)) - 1)

tan(π/12) = √(2 / (1 + √3 / 2) - 1)

tan(π/12) = √(4 / (2 + √3) - 1)

tan(π/12) = √(4 / (2 + √3) - (2 + √3) / (2 + √3))

tan(π/12) = √((2 - √3) / (2 + √3))

tan(π/12) = √(7 - 4 √3)

a, b, c, d are correct.

step-by-step explanation:

we have g(x) = 3 and are required to find which option is a translation of g(x).

now, 'translation' is a transformation which means to shift the function in any direction and the general form of translation is g(x) + k, where k is any real number.

so, according to the options we see that all are correct as they satisfy the general form of translation.

also, it can be seen from the graph below that the function g(x) = 3 is shifted from its position in all the four options.