Question 1158028
The standard position of an angle is vertex at the origin, initial side along the positive x-axis.
We measure angles counterclockwise, and could define an angle measure as negative or greater than 360 degree,
but the trigonometric functions only care about the position of the terminal side.
I will assume {{{0^o<=theta<360^o}}} for all angles.
 
Terminal side passing through {{{P(-8,6)}}} :
{{{drawing(300,300,-10,2,-2,10,
grid(0),circle(-8,6,0.1),circle(-0.8,0.6,0.1),
triangle(0,0,-8,6,-8,0),locate(-8,6.8,P),
triangle(0,0,-0.8,0.6,-0.8,0),locate(-1.4,0.9,A),
red(circle(0,0,1)),blue(arrow(0,0,2,0)),
blue(arrow(0,0,-9.6,7.2)),locate(0.1,0.6,O),
green(arc(0,0,3.6,3.6,-143.1,0)),
green(triangle(-1.44,1.08,-1.44,1.38,-1.14,1.08)),
locate(1,2,green(theta)),
red(arc(0,0,6,6,-180,-143.1)),
locate(-2.8,1.1,red(alpha))
)}}} There is a large right triangles with hypotenuse {{{OP}}} and a similar triangle whose hypotenuse, {{{OA}}} , is a radius of the unit circle shown in red. 
The large one has legs of length {{{8}}} and {{{6}}} , and hypotenuse {{{OP=sqrt(6^2+8^2)=10}}} . The length of the small right triangle's hypotenuse, {{{OA=1}}}} ,is 10 times smaller, and so are the legs.
The function {{{sin(green(theta))}}} is defined as the y-coordinate of point A, {{{sin(green(theta))=highlight(0.6)}}} , and {{{cos(green(theta))}}} is defined as the x-coordinate of point A, {{{cos(green(theta))=-0.8}}} .
We can calculate {{{sin(red(alpha))=6/10=0.6}}} as a trigonometric ratio. and determine that {{{red(alpha)=36.9^o}}}.
{{{green(theta)=180^o-36.9^o=143.1^o}}}
Trigonometric cosine and sine functions of {{{green(theta)}}} , defined as the x-coordinate and the y-coordinate of point A respectively are numerically the same as those for {{{red(alpha)}}} , but may be positive or negative depending on the quadrant.
The sign will be the same for coordinates of any point on the terminal side.
 
Terminal side passing through {{{P(2,-1)}}} :
{{{P(2,-1)}}} is in quadrant IV, with positive x-coordinate, so {{{cos(theta)=cos(alpha)>0}}}.
The large right triangle, in this case, has leg lengths of 2, and 1, and a hypotenuse length of {{{OP=sqrt(2^2+1^3)=sqrt(4+1)=sqrt(5)}}} .
{{{cos(theta)=cos(alpha)=2/sqrt(5)}}} and {{{sec(theta)=1/cos(theta)=highlight(sqrt(5)/2=1.118)}}}(rounded)
The angles involved would be {{{alpha=26.56^o}}} , {{{theta=360^o-26.56^o=333.44^o}}}
 
Terminal side passing through {{{P(0,1)}}} :
P is on the unit circle, so its x-coordinate and y-coordinate are {{{cos(theta)}}} and {{{sin(theta)}}} respectively.
{{{cos(theta)=0}}} , {{{sin(theta)=1}}} and {{{tan(theta)=sin(theta)/cos(theta)}}} is undefined.
  
Terminal side passing through {{{P(-4,-2)}}} :
Booth coordinates are negative, and so will be sine and cosine.
OP would be the hypotenuse of a right triangle with leg lengths 4, and 2.
The hypotenuse length is {{{OP=sqrt(4^2+2^2)=sqrt(16+4)=sqrt(20).
{{{cos(theta)="x-coordinate"/OP=-4/sqrt(20)=-0.894}}}