document.write( "Question 167514This question is from textbook Functions and Relations
\n" ); document.write( ": The 45degree 60degree 75degree triangle has angles in arithmetic sequence.
\n" ); document.write( "In triangle ABC, AB=1, angle A=45degrees, and angle B=60 degrees, without using a calculator how can I show these expressions?
\n" ); document.write( "a)the lengths of AC and BC
\n" ); document.write( "b)sin 75degrees, cos 75degrees, tan75degrees
\n" ); document.write( "c)sin 15degrees, cos 15degrees, tan 15degrees \n" ); document.write( "
Algebra.Com's Answer #123515 by jojo14344(1513)\"\" \"About 
You can put this solution on YOUR website!
I'll leave you the sketch, hopefully we can follow together.
\n" ); document.write( "In the triangle ABC, given:
\n" ); document.write( "\"A=45%5Eo\"; \"B=60%5Eo\"; \"C=75%5Eo\"
\n" ); document.write( "\"AB=1unit\"
\n" ); document.write( "Pardon me for the illustration below is just for reference --> NOT TO SCALE. So it will be easy for us to go along.
\n" ); document.write( " ---> BLUE circle>>>\"A=45%5Eo\"; GREEN Circle>>> \"B=60%5Eo\"; RED Circle>>> \"C=75%5Eo\". Also,BLUE LINE =\"AC\"; GREEN LINE=\"AB\"; RED LINE=\"BC\"\r
\n" ); document.write( "\n" ); document.write( "a)the lengths of AC and BC
\n" ); document.write( "In order to find \"AC\" and \"BC\" with given data,w e draw a line from \"B\" to line \"AC\" and mark LINE \"BD\". See below:
\n" ); document.write( " ---> BLACK Line=\"BD\"
\n" ); document.write( "We know \"AC=AD%2BDC\" -------------> Eqn 1 >>>> as shown on the graph
\n" ); document.write( "We find \"ABD\" forms right triangle. By Trigo Functions:
\n" ); document.write( "Find first Line \"AD\" ---> \"tan45%5Eo=opp%2Fadj=AD%2FBD\"--->\"AD=tan45%5Eo%28BD%29\"---------> Eqn 2:\"BD=unknown\"
\n" ); document.write( "But, \"sin45%5Eo=BD%2FAB\" --> \"BD=sin45%5Eo%28AB%29\"--------------> Eqn 3
\n" ); document.write( "Subst. Eqn 3 in Eqn 2:
\n" ); document.write( "\"AD=%28tan45%5Eo%29%28sin45%5Eo%29%28AB%29\" ------------------> Eqn 4
\n" ); document.write( ".
\n" ); document.write( "Next, Line \"DC\", On Angle \"C=75%5Eo\":
\n" ); document.write( "\"tan75%5Eo=opp%2Fadj=BD%2FDC\"
\n" ); document.write( "\"DC=BD%2F%28tan75%5Eo%29=Eqn3%2F%28tan75%5Eo%29=sin45%5Eo%28AB%29%2F%28tan75%5Eo%29\" -----------> Eqn 5
\n" ); document.write( "Therefore, Via Eqn 1:
\n" ); document.write( "------------------> Eqn 6
\n" ); document.write( "*Note: \"AB=1\" as per given
\n" ); document.write( ".
\n" ); document.write( "Next, Line \"BC\"=?
\n" ); document.write( "\"cos75%5Eo=adj%2Fhyp=DC%2FBC\"
\n" ); document.write( "\"BC=DC%2Fcos75%5Eo=Eqn.5%2Fcos75%5Eo\"
\n" ); document.write( "\"BC=%28%28sin45%5Eo%29%28AB%29%2Ftan75%5Eo%29%29%2Fcos75%5Eo\"
\n" ); document.write( "\"highlight%28BC=%28sin45%5Eo%29%28AB%29%2F%28tan75%5Eo%29%28cos75%5Eo%29%29\" ---------------> Eqn 7
\n" ); document.write( ".
\n" ); document.write( "b)sin 75degrees, cos 75degrees, tan75degrees
\n" ); document.write( "First \"sin75%5Eo\"=?,
\n" ); document.write( "On Right Triangle \"CBD\" with \"C=75%5Eo\":
\n" ); document.write( "\"sin75%5Eo=opp%2Fhyp=BD%2FBC=Eqn.3%2FEqn.7\"
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\"highlight%28sin75%5Eo=%28tan75%5Eo%29%28cos75%5Eo%29%29\"
\n" ); document.write( "Next, \"cos75%5Eo\"=?,
\n" ); document.write( "\"cos75%5Eo=adj%2Fhyp=DC%2FBC=Eqn.5%2FEqn.7\"
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\"highlight%28cos75%5Eo=cos75%5Eo%29\"
\n" ); document.write( ".
\n" ); document.write( "Next, \"tan75%5Eo\"=?
\n" ); document.write( "\"tan75%5Eo=opp%2Fadj=BD%2FDC=Eqn.3%2FEqn.5\"
\n" ); document.write( "\"tan75%5Eo=%28%28sin45%5Eo%29%28AB%29%29%2F%28%28sin45%5Eo%29%28AB%29%2Ftan75%5Eo%29\"
\n" ); document.write( "
\n" ); document.write( "\"highlight%28tan75%5E0=tan75%5Eo%29\"
\n" ); document.write( ".
\n" ); document.write( "c)sin 15degrees, cos 15degrees, tan 15degrees
\n" ); document.write( "We draw a line from \"A\" to \"BD\" as shown:
\n" ); document.write( "Mark Line \"AE\".See below:
\n" ); document.write( " ----> We call it angle \"A%5B1%5D=15%5Eo\" from A --> E.
\n" ); document.write( ".
\n" ); document.write( "We solve for \"AE\":
\n" ); document.write( "\"highlight%28cos15%5Eo=adj%2Fhyp=AD%2FAE%29\"
\n" ); document.write( "\"highlight%28AE=AD%2Fcos15%5Eo=Eqn.4%2Fcos15%5Eo%29\" ------------> Eqn 8
\n" ); document.write( "Next, \"sin15%5Eo\"=?
\n" ); document.write( "\"highlight%28sin15%5Eo=opp%2Fhyp=ED%2FAE=ED%2FEqn.8%29\"
\n" ); document.write( "\"highlight%28ED=%28sin15%5Eo%29%28Eqn.8%29%29\" -------------------> Eqn 9
\n" ); document.write( "Next, \"tan15%5Eo\"=?
\n" ); document.write( "\"highlight%28tan15%5Eo=opp%2Fadj=AD%2FED=Eqn.4%2FEqn.9%29\"
\n" ); document.write( "Try to finish it off with what we've computed on the above EQUATIONS 1-9:
\n" ); document.write( ".
\n" ); document.write( "Thank you,
\n" ); document.write( "Jojo \n" ); document.write( "
\n" );