tan75\2
let's first handle 75 degrees
tan75 = tan(30+45) , skipping the degree symbols
= (tan30 + tan45)/(1 - tan30tan45)
= (1/sqr3 + 1)/(1 - (1/sqr3)(1))
= (sqr3 + 1)/(sqr3 - 1)
tan 75 = 2tan(75/2)/( 1 - tan^2 (75/2) )
let tan(75/2) = x for easier typing
(sqr3 + 1)/(sqr3 - 1) = 2x/(1 - x^2)
argghh, I used the quadratic formula and got
x = (2(sqr( 2 + sqr3) ) - 1 )/(2 + sqr3)
= appr .7673
15-8√3
√2+√3
To find the value of tan(75°/2), you can use the half-angle formula for tangent:
tan(x/2) = ±√[(1 - cosx) / (1 + cosx)]
In this case, x = 75°, so we want to find:
tan(75°/2) = ±√[(1 - cos75°) / (1 + cos75°)]
First, let's find cos75°. We can use the sum-to-product formula for cosine:
cos(A + B) = cosA * cosB - sinA * sinB
cos75° = cos(45° + 30°) = cos45° * cos30° - sin45° * sin30°
cos45° = √(2) / 2
cos30° = √(3) / 2
sin45° = √(2) / 2
sin30° = 1 / 2
cos75° = (√2 / 2) * (√3 / 2) - (√2 / 2) * (1 / 2)
cos75° = (√6 / 4) - (√2 / 4)
cos75° = (√6 - √2) / 4
Now we can substitute this value into the half-angle formula:
tan(75°/2) = ±√[(1 - (√6 - √2) / 4) / (1 + (√6 - √2) / 4)]
Simplifying further, we get:
tan(75°/2) = ±√[(4 - √6 + √2) / (4 + √6 - √2)]
This gives us the value of tan(75°/2) with the ± sign indicating the positive and negative possibilities.