Question

What is the exact value of tan 75?

Answer 1

2 + √3.

Explanation:

tan 75 = tan (45 + 30)

= (tan 45 + tan 30) / (1 - tan 45 tan 30)

= ( 1 + 1/√3) / (1 - 1*1/√3)

= (1 + √3) / √3 * √3 / (√3 - 1)

= √3 + 1/ √3 - 1

Multiply top and bottom by √3 + 1;

= (√3 + 1)(√3 + 1) / (√3 - 1)(√3 + 1)

= 3 + 2√3 + 1 / (3 - 1)

= (4 + 2√3) / 2

= 2 + √3.