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Geometry Level 3

Find the value of $x$ between 0 and 180 such that

$\tan({ 120 }^{ \circ }-x^{ \circ })=\frac { \sin{ 120 }^{ \circ }-\sin x^{ \circ } }{ \cos{ 120 }^{ \circ }-\cos x^{ \circ }}.$

100 No such

x

110 90

2 solutions

John Aries Sarza
May 24, 2014

$tan({ 120 }^{ o }-x)=\quad \frac { 2cos\frac { { 120 }^{ o }+x }{ 2 } sin\frac { { 120 }^{ o }-x }{ 2 } }{ -2sin\frac { { 120 }^{ o }+x }{ 2 } sin\frac { { 120 }^{ o }-x }{ 2 } } =\quad -cot\frac { { 120 }^{ o }+x }{ 2 } \\ \quad \quad \quad \quad \quad =\quad -tan({ 90 }^{ o }-\frac { { 120 }^{ o }+x }{ 2 } )=\quad tan(\frac { x-60^{ o } }{ 2 } )$ $tan({ 120 }^{ o }-x)=\quad tan(\frac { x-60^{ o } }{ 2 } )$ ${ 120 }^{ o }-x=\quad \frac { x-60^{ o } }{ 2 } \\ \boxed{x={ 100 }^{ 0 }}\\$

Done same way nice solution

Mardokay Mosazghi - 7 years ago

wow..didnt even thought of that.

pranav jangir - 7 years ago

Nice solution, but please tell me only one thing. How did $-cot\frac { 120°+x }{ 2 }$ become $-tan(90°-\frac { 120°+x }{ 2 } )$ ? Any formula?

Arsalan Iqbal - 7 years ago

If we have a right triangle ABC at c=90, $cot\quad A=\quad tan\quad B\quad =tan(90-A)$

John Aries Sarza - 7 years ago

Arpan Mathur
Feb 7, 2017

120 is another answer for x.

As the denominator can't be zero, 120 isn't in the domain.

Fatemeh Jafari - 3 years ago

The question must have been edited before your comment.

Arpan Mathur - 2 years, 11 months ago

You're Just My X

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