given that tan 75 2+root3,without using a table or calculator, determine 5tan15 leaving your answer in the form a+b root c

Answer 1

To find 5tan(15), we can use the half-angle identity for tangent:

tan(x/2) = sqrt((1-cos(x))/(1+cos(x)))

Since x = 30 degrees, we have:

tan(15) = sqrt((1-cos(30))/(1+cos(30)))

To simplify, let's use the trigonometric identities:

cos(30) = sqrt(3)/2

Now we can substitute this value into our equation:

tan(15) = sqrt((1 - sqrt(3)/2)/(1 + sqrt(3)/2))

To rationalize the denominator, multiply both the numerator and denominator by the conjugate of the denominator:

tan(15) = sqrt((1 - sqrt(3)/2)/(1 + sqrt(3)/2)) * (sqrt(3)/sqrt(3))

Simplifying, we have:

tan(15) = sqrt((sqrt(3) - 3)/(sqrt(3) + 3))

To get the answer in the form a + b√c, we can multiply the numerator and denominator by the conjugate of the numerator:

tan(15) = (sqrt(3) - 3)/(sqrt(3) + 3) * (sqrt(3) - 3)/(sqrt(3) - 3)

Expanding and simplifying, we have:

tan(15) = (3 - 2sqrt(3))

So, 5tan(15) = 5(3 - 2sqrt(3)) = 15 - 10sqrt(3)

Thus, the answer is 15 - 10√3.