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

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Answer 1 · 2014-10-24T04:37:38

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

We leverage that information to evaluate tan(105).

$tan(a+b)=(tan(a)+tan(b))/(1-tan(a)tan(b))$

$tan(60+45)=(tan(60)+tan(45))/(1-tan(60)tan(45))$

$tan(105)=(tan(60)+tan(45))/(1-tan(60)tan(45))$

$tan(105)=(sqrt(3)+1)/(1-sqrt(3)*1)$

$tan(105)=(sqrt(3)+1)/(1-sqrt(3))$

Rationalize

$tan(105)=(sqrt(3)+1)/(1-sqrt(3))*(1+sqrt(3))/(1+sqrt(3))$

$tan(105)=(sqrt(3)+3+1+sqrt(3))/(1-3)$

$tan(105)=(2sqrt(3)+4)/(-2)$

$tan(105)=-3.732050808$

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

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