Question 197567
find the cosine of the angle formed with the coordinates (-6, 7) located in standard position
<pre><font size = 4 color = "indigo"><b>
Plot the point (-6,7)

{{{drawing(400,400,-10,10,-10,10, graph(400,400,-10,10,-10,10),
line(-5.9,7,-6.1,7), line(-6,6.9,-6,7.1), line(-5.9,7.1,-6.1,6.9), line(-6.1,7.1,-5.9,6.9), locate(-7,8,"(-6,7)") )}}}

Draw the radius vector, r, from the origin (0,0) to the point (-6,7). 

{{{drawing(400,400,-10,10,-10,10, graph(400,400,-10,10,-10,10),
line(-5.9,7,-6.1,7), line(-6,6.9,-6,7.1), line(-5.9,7.1,-6.1,6.9), line(-6.1,7.1,-5.9,6.9), locate(-7,8,"(-6,7)"), line(0,0,-6,7),
locate(-3,4.5,r) )}}}

Indicate the angle.  I'll call it {{{alpha}}}.

{{{drawing(400,400,-10,10,-10,10, graph(400,400,-10,10,-10,10),
line(-5.9,7,-6.1,7), line(-6,6.9,-6,7.1), line(-5.9,7.1,-6.1,6.9), line(-6.1,7.1,-5.9,6.9), locate(-7,8,"(-6,7)"), line(0,0,-6,7),
blue(arc(0,0,2,-2,0,139.4) ), locate(.5,1.5,alpha), locate(-3,4.5,r)
 )}}}

Draw a vertical line perpendicular down to the x-axis, and
label it {{{y=7}}}.

{{{drawing(400,400,-10,10,-10,10, graph(400,400,-10,10,-10,10),
line(-5.9,7,-6.1,7), line(-6,6.9,-6,7.1), line(-5.9,7.1,-6.1,6.9), line(-6.1,7.1,-5.9,6.9), locate(-7,8,"(-6,7)"), line(0,0,-6,7),
blue(arc(0,0,2,-2,0,139.4) ), locate(.5,1.5,alpha), line(-6,7,-6,0),
locate(-7.3,3.5,"y=7"), locate(-3,4.5,r)
 )}}}

Draw a horizontal line from the origin to the bottom of
that line, completing a right triangle.  Label it {{{x=-6}}}

{{{drawing(400,400,-10,10,-10,10, graph(400,400,-10,10,-10,10),
line(-5.9,7,-6.1,7), line(-6,6.9,-6,7.1), line(-5.9,7.1,-6.1,6.9), line(-6.1,7.1,-5.9,6.9), locate(-7,8,"(-6,7)"), line(0,0,-6,7),
blue(arc(0,0,2,-2,0,139.4) ), locate(.5,1.5,alpha), line(-6,7,-6,0),
locate(-7.3,3.5,"y=7"), locate(-3,4.5,r), line(0,0,-6,0), 
locate(-4,1,"x=-6")
 )}}}

Now we find the length of {{{r}}} by the Pythagorean theorem:

{{{r^2=x^2+y^2}}}
{{{r^2=(-6)^2+(7)^2}}}
{{{r^2=36+49}}}
{{{r^2=85}}}
{{{r=sqrt(85)}}}

So we put {{{r=sqrt(85)}}} for {{{r}}}: 

{{{drawing(400,400,-10,10,-10,10, graph(400,400,-10,10,-10,10),
line(-5.9,7,-6.1,7), line(-6,6.9,-6,7.1), line(-5.9,7.1,-6.1,6.9), line(-6.1,7.1,-5.9,6.9), locate(-7,8,"(-6,7)"), line(0,0,-6,7),
blue(arc(0,0,2,-2,0,139.4) ), locate(.5,1.5,alpha), line(-6,7,-6,0),
locate(-7.3,3.5,"y=7"), locate(-3,4.5,r=sqrt(85)), line(0,0,-6,0), 
locate(-4,1,"x=-6")
 )}}}

Now since

{{{cos(alpha)=x/r}}}

{{{cos(alpha)=(-6)/sqrt(85)}}}

{{{cos(alpha)=-6/sqrt(85)}}}

Edwin</pre>