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avatar+54 

\(F\)Let \(\alpha\) and \(\beta\) be acute angles whose sum is also acute, such that \(\angle BAC = \alpha, \angle DAB = \beta\) and \(AB = 1\).Let \(E\) be the foot of the perpendicular from \(D \) to \(AC\), and  be the foot of the perpendicular from \(B \) to \(DE\), as shown below:

a) Prove that \(\triangle DBF\) is similar to \(\triangle ABC\).

b) Calculate \(AC, BC, AD, BD, BF\) and \(DF\) in terms of trigonometric functions of  \(\alpha\)and \(\beta\).

c) Use the diagram above to provide another proof of the sine and cosine angle sum identities for acute angles  \(\alpha\)and \(\beta\)whose sum is below\(90^{\circ}\) .

 Jul 25, 2021
 #1
avatar+54 
+2

Sorry, I meant that F be the foot of the perpendicular, i had no idea how the f went to the front 

 Jul 25, 2021
 #2
avatar+121056 
+2

I can do a few

 

(a)  Angles   BCA   and  DFB     = 90

      Angle DBA   = angle  FBC  = 90

Angle DBA   =Angle  DBF  +  Angle FBA

Angle FBC  =  Angle ABC  +  Angle  FBA

Which implies  that angles  DBF  and ABC  are equal

Thus.....by AA congruence.....triangle ABC  is similar to triangle DBF

 

 

cool cool cool  

 Jul 25, 2021
 #3
avatar+121056 
+2

(b)

 

cos  a   =  AC  /1    ⇒      AC   =cos   a

 

sin a  = BC  /  1    ⇒   BC   =   sin a

 

cos b   =  1 / AD   ⇒   AD  =  1/ cos b  ⇒   AD  =  sec  b

 

tan b =   BD  / 1     ⇒     BD  = tan  b

 

Not sure  about  the last  two.......but here's  what I  get

 

angle  a  =   angle  FDB .......so  

 

sin  a   =   sin  FDB

 

sin a   =    BF  / DB      

 

sin a  =  BF  /  tanb    ⇒     sin a tan b   =  BF  

 

Similarly

 

cos  a  =  cos   FDB

 

cos  a   =   DF /  DB    

 

cos  a  =  DF / tan  b   ⇒   cos a  tan b   =    DF

 

 

cool cool cool

 Jul 25, 2021
 #5
avatar+54 
+2

TYSM!

I was just trying to figure them out, but i couldnt.

Thanks! :)

elloooo  Jul 25, 2021
 #6
avatar+121056 
+2

Ur  Welcome   !!!!!

 

cool cool cool

CPhill  Jul 25, 2021
 #4
avatar+121056 
+2

For  the  last one......see if this helps  :    https://themathpage.com/aTrig/sum-proof.htm

 

 

 

cool cool cool

 Jul 25, 2021

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