What is the exact value of tan 105 degrees?

Answer 1

To find the exact value of tan 105°.

First, of all, we note that

sin 105° = cos 15°; and

cos 105° = -sin 15°.

Thus, tan 105° = -cot 15° = -1 / tan 15°.

Using the formula

tan(α - β) = (tan α - tan β) / (1 + tan α tan β);

and using, also, the familiar values

tan 45° = 1, and

tan 30° = ½ / (½√3) = 1/√3 = ⅓√3;

we have,

tan 15° = (1 - ⅓√3) / (1 + ⅓√3);

whence,

cot 15° = (1 + ⅓√3) / (1 - ⅓√3)

= (√3 + 1) / (√3 - 1) {multiplying through by √3}

= (√3 + 1)2 / (√3 + 1)(√3 - 1)

= (3 + 2√3 + 1) / (3 - 1)

= (4 + 2√3) / 2

= 2 + √3.

Therefore,

tan 105° = -cot 15° = -2 - √3,

which is the result we sought.

We are asked the exact value of tan 105°, which we gave above.

We can test the above result to 9 decimal places, say, by means of a calculator:

-2 - √3 = -3.732050808; and

tan 105° = -3.732050808;

thus indicating that we have probably got the right result.