Question

Tan37.5

Answer 1

The angles 37, 52, and 90 correspond to a 3,4,5 sided right angled triangle.

Using that information, you can take tan37.5 = 3/4

Use a calculator for a more accurate value.

Answer 2

this is probably a question asking for EXACT values

37.5 = 1/2 of 75
and 75 = 30+45

so find tan75 = tan(30+45)
= (tan30 + tan45)/( 1 - tan30*tan45)
= (1/√3 + )/(1 - 1/√3)
= (1+√3)(√3 - 1)

recall tan 2A = 2tan A/(1 - tan^2 A)

tan 75 = 2tan37.5/(1-tan^2 37.5)

let tan37.5 = x
(1+√3)(√3 - 1) = 2x/(1-x^2)
(1+√3)(1-x^2) = 2x(√3 - 1)

expand, collect like terms and you will have a quadratic.
Use the quadratic formula to find the exact value.
I know it looks messy but stick to it

Answer 3

use the addition formula as above to find
sin(75) = sin45*cos30+cos45*sin30 = (√3-1)/√8
cos(45) = cos45*cos30-sin45*sin30 = (√3-1)/√8

Now using the half-angle formula,

tan(37.5) = (1-cos75)/sin75 = sin75/(1+cos75)

No quadratics to solve, but you still have messy algebra.