How do you use the sum and difference identities to find the exact value of tan 105 degrees?

Question

20862 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2014-10-24T04:37:38

The special triangles, $30 : 60 : 90 and 45 : 45 : 90$ $30:60:90 and 45:45:90$ , allow us to evaluate sine and cosine and tangent.

We leverage that information to evaluate tan(105).

$tan (a + b) = \frac{tan (a) + tan (b)}{1 - tan (a) tan (b)}$ $tan(a+b)=(tan(a)+tan(b))/(1-tan(a)tan(b))$

$tan (60 + 45) = \frac{tan (60) + tan (45)}{1 - tan (60) tan (45)}$ $tan(60+45)=(tan(60)+tan(45))/(1-tan(60)tan(45))$

$tan (105) = \frac{tan (60) + tan (45)}{1 - tan (60) tan (45)}$ $tan(105)=(tan(60)+tan(45))/(1-tan(60)tan(45))$

$tan (105) = \frac{\sqrt{3} + 1}{1 - \sqrt{3} \cdot 1}$ $tan(105)=(sqrt(3)+1)/(1-sqrt(3)*1)$

$tan (105) = \frac{\sqrt{3} + 1}{1 - \sqrt{3}}$ $tan(105)=(sqrt(3)+1)/(1-sqrt(3))$

Rationalize

$tan (105) = \frac{\sqrt{3} + 1}{1 - \sqrt{3}} \cdot \frac{1 + \sqrt{3}}{1 + \sqrt{3}}$ $tan(105)=(sqrt(3)+1)/(1-sqrt(3))*(1+sqrt(3))/(1+sqrt(3))$

$tan (105) = \frac{\sqrt{3} + 3 + 1 + \sqrt{3}}{1 - 3}$ $tan(105)=(sqrt(3)+3+1+sqrt(3))/(1-3)$

$tan (105) = \frac{2 \sqrt{3} + 4}{- 2}$ $tan(105)=(2sqrt(3)+4)/(-2)$

$tan (105) = - 3.732050808$ $tan(105)=-3.732050808$

To evaluate make sure that the calculator is in Degree mode.

Normal float auto real degree MP math formula