7 Hard Trig Problems

7 Hard Trig Problems-Can You Solve These Challenging Gemetry Problems? Updated on June 7, 2016 Trignmry is  r f gmry h ls wih lghs  gs f ringls. A sc my w-msnl fgrs  hr-msnl sl 9

shs c b brk w  rigls rgmry s ls s  slv my vnc rblms nvlvg mr cmlic shs. Trigmrc rlns  fcs r bs  h rls bw h ss f rgh ringls. Th si f  nn-righ gl s h r f h s s  h hys. Th cs is h ri f h jc s  h hys. A filly h g is h ri f h s s  h jcn s. My l s h mnmc "ShCh"  rmmbr hs rs. Try hs sv hr gmry rblms  s yr sklls  sig rg  f ngls  lghs  rgls.

(1) Right Triangles Triangles of Equal Equal Perimeter Perimeter

Problem: Th w rgh ringls shw bv hv ql rmrs. Th hys f h rng rigl s n lg f h grn rgl sck  

f . If h smlls gl f h rg rgls s  grs wh r h ngls f h grn rgh rngl? Solution: W '   kw h si lghs f h rgls  slv fr h nkw gls ly hir rlv lnghs r ncssry. Fr h sk f smlciy l h w lgs f h rg rgl b cs() n s() n l h hys ql . Th gr rngl h hs  lg f lgh  hr lngh f knwn lgh X n  hys f sqr(X +  ).

Sinc h w rgls' rmrs r ql w hv h qi s( + cs( +   + X + sqr(X + ) s() + cs() X + sqr(X + )

=



co(20)

If we use the technique for solvng algebraic equations wit square root and liear terms we get [sin(20)

+

cos(20) - XJA2

=

X A2

si(20)A2 + 2sin(20)cos(20) A = X  2 + 1 1 

+

+

1

2cos(20)A2 - 2sin(20)X - cos(20)X

2sn(20)cos(20) - 2sn(20)X - 2cos(20)X X  2 + 1 +

+

+

X A2

X A2

A

2sn(20)cos(20) - 2sin(20)X - 2cos(20) = O X  sin(20)cos(20)/[sin(20 =

+

cos(20)] = .250753

Te angles of the green trangle are 90 deges, arctan(X/1)" 14.08 degrees, and arctan(1/X)" 75.92 degees

(2) Find the Missing Angles

Problem: Consider a triangle wth one angle equal to 45 degrees An altitude is drawn to partiton the original trangle into two smaller right triangles, colored yellow and purple n the diagram above If the area of the purple triangle is 15 times the area of the yellow triangle what are the angles of the orginal whole trangle? Solution:

An altitude lne splits the triangle nto two right trangles The yellow trangle s a 45-45-90-degree triangle so its legs have equal length Without any loss of generalit, we can set that lengt equal to 1 If the purple triangle has 15 times the area then its leg lengths are 1 and 15 Call the top unknown angle of the prple triangle x degrees and the bottom unknown angle (90-x) degrees. From trigonometry, we kow that tan(x) x

=

=

151

=

15

arctan(15)" 5631 degrees

Snce 90 - 5631 3369 and 45 + 5631 10131 we now have all the angles of the orignal triangle: 45  33.69  and 101.31  =

=

°

°

(3) Isosceles Integer Triangle

°

si() = (34 I R   M th fe g   l e = 23  1 •1-.

3R 4

(8) Mmu Aa o a Tg

:  s   t h gth  a 7 V  l e ng o the th se   t the   e ng l d? S   g, th  m g Hm li on  l   wh    n goo   ll  i   th ' F tl  8 e r   ng i 8    b th 

O

A 05�" - X - 6  =

 pt   x   y   d     l  z   r       gn k  l pb ll     ,  n   idN   P  Q         • 05)

17

b

    =  a  = , e n le quan In l equa l e  , w  ge f O des   dage Te  I  (3(      n n&      v f )     9 g r & • .

2

     the tg  a t ag  leg ar  a  a w hte   T  ng de g u be     ch   re  r  l agle  .  3 .. 



 Rght   Equl A



b

(