Question 207450
Find the exact value of tan(15°) without using tables or a calculator
<pre><font size =4 color = "indigo"><b>
{{{"15°"}}} is not a special angle, however {{{"45°"}}} and {{{"30°"}}}
are.  And {{{"15°"="45°"-"30°"}}}, so we can use this formula:

{{{tan(alpha-beta)=(tan(alpha)-tan(beta))/(1+tan(alpha)tan(beta))}}}

where {{{alpha="45°"}}} and {{{beta="30°"}}}:

{{{tan("15°")=tan("45°"-"30°")= (tan("45°")-tan("30°"))/(1+tan("45°")tan("30°"))=(1-sqrt(3)/3)/(1+(1)(sqrt(3)/3))=(1-sqrt(3)/3)/(1+sqrt(3)/3)}}}

Multiply top and bottom by {{{3}}}

{{{3(1-sqrt(3)/3)/(3(1+sqrt(3)/3))=(3-sqrt(3))/(3+sqrt(3))}}}

Rationalize the denominator:

{{{(  (3-sqrt(3))(3-sqrt(3))   )/(  (3+sqrt(3))(3-sqrt(3))  )

= (9-3sqrt(3)-3sqrt(3)+(sqrt(3))^2)/(9-3sqrt(3)+3sqrt(3)-(sqrt(3))^2)=

 (9-3sqrt(3)-3sqrt(3)+(sqrt(3))^2)/(9-cross(3sqrt(3))+cross(3sqrt(3))-(sqrt(3))^2)}}}=

{{{(9-6sqrt(3)+3)/(9-3)=(12-6sqrt(3))/6= (6(2-sqrt(3)))/6=

 (cross(6)(2-sqrt(3)))/cross(6)= 2-sqrt(3)}}}.

Check by using your calculator.  

{{{tan("15°")= .2679491924}}}

Then find the decimal value for

{{{2-sqrt(3) = .267941924}}}

So it checks.

So the answer is 

{{{tan("15°")=2-sqrt(3)}}}

Edwin</pre>