Question 837932
If the only length you have is the length of one leg, but you know the measures of the angles, you need to use the trigonometric ratios:
{{{tangent=opposite_side/adjacent_side}}}
{{{cosine=adjacent_side/hypotenuse}}}
 
Where is the side with length 8?
Is it adjacent to the {{{23^o}}} angle or to the {{{67^o}}} angle?
 
ADJACENT TO THE {{{23^o}}} ANGLE:
{{{drawing(400,200,-1,9,-1,4,
triangle(0,0,8,0,8,3.4),rectangle(8,0,7.8,0.2),
locate(1,0.6,37^o),locate(4,0,8),
locate(8.1,2,leg),locate(4,1.7,h)
)}}} {{{tan(23^o)=leg/8}}} and {{{cos(23^o)=8/h}}}
{{{tan(23^o)=0.424475}}} (rounded) and {{{cos(23^o)=0.920505}}} (rounded)
{{{tan(23^o)=leg/8}}} --> {{{leg=8*tan(23^o)=3.40}}} (rounded)
{{{cos(23^o)=8/h}}} --> {{{h=8/cos(23^o)=8.69}}} (rounded)
 
ADJACENT TO THE {{{67^o}}} ANGLE:
{{{drawing(400,200,-0.5,19.5,-1,9,
triangle(0,0,18.85,0,18.85,8),rectangle(18.85,0,18.35,0.5),
locate(18.9,4.5,8),locate(17.7,7.8,67^o),
locate(9,0,leg),locate(9.4,4,h)
)}}} {{{tan(67^o)=leg/8}}} and {{{cos(67^o)=8/h}}}
{{{tan(67^o)=2.355852}}} (rounded) and {{{cos(67^o)=0.390731}}} (rounded)
{{{tan(67^o)=leg/8}}} --> {{{leg=8*tan(67^o)=18.85}}} (rounded)
{{{cos(67^o)=8/h}}} --> {{{h=8/cos(67^o)=20.47}}} (rounded)