Question 1195236
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Given that 0 < x < 90° and sin(x+60°) = cos(2x), find the exact value of tan(x+20°)
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<pre>
Your starting equation is

    sin(x+60°) =cos(2x).         (1)


Since  cos(2x) = sin(90°-2x), you can re-write equation (2) in the next equivalent form

    sin(x+60°) = sin(90°-2x).    (2)


Since both angles, x+60° and 90°-2x are acute, equation (2) implies

    x+60° = 90°-2x.


Solve it and find x

    x + 2x = 90° - 60°,

      3x   =    30°,

       x   =    10°.


THEREFORE,  tan(x+20°) = tan(10°+20°) = tan(30°) = {{{sqrt(3)/3}}}.


<U>ANSWER</U>.  If  0 < x < 90°  and  sin(x+60)° = cos(2x),  then  tan(x+20°) = {{{sqrt(3)/3}}}.
</pre>

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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Found without using a calculator or tables, but exclusively 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;using the power of knowledge and intellect.


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But notice that, &nbsp;without using a calculator or tables, &nbsp;I found &nbsp;EXACT &nbsp;EXPRESSION 

for &nbsp;tan(x+20°), &nbsp;ONLY - - - but not the value of &nbsp;tan(x+20°).



It is &nbsp;POSSIBLE &nbsp;to find exact expression without using calculator or tables,
but it is &nbsp;IMPOSSIBLE &nbsp;to find the &nbsp;VALUE &nbsp;without using a calculator or a table.


THEREFORE, &nbsp;the problem's formulation itself &nbsp;IS &nbsp;NOT &nbsp;CORRECT.


The correct formulation should ask to find &nbsp;EXACT &nbsp;EXPRESSION &nbsp;for &nbsp;tan(x+20°), &nbsp;but not the exact value of &nbsp;tan(x+20°).



Today, &nbsp;I saw &nbsp;SEVERAL &nbsp;similar posts from you for other expressions, &nbsp;also asking 
to find exact value without a calculator or a table.


All these formulations are &nbsp;INCORRECT &nbsp;due to the same reason.
Correct formulations should ask to find exact expressions, &nbsp;but not about exact values.