Question 662885
I could try to explain it for a 5-year old who reads very well, and can do basic arithmetic. If you know of any such a 5-year old, I'd love to try an explanation on them for a laugh.
In the meantime, I'll try an explanation that I believe might work for you. If it doesn't you can let me know via the website in a "thank you" note. I would answer.
 
EXPLANATION FOR YOU:
When you have a set of coordinates, like the one below, the axes divide it into four {{{90^o}}} angles called quadrants.
The quadrants are named as the first, second, third, and fourth quadrants.
They use roman numerals I, II, II, and IV to label them.
They start at the right end of the x-axis and name them in order going counterclockwise, like this:
{{{drawing(300,300,-5,5,-5,5,
grid(0),
locate(2,3,I),locate(-3,3,II),
locate(-3,-2,III),locate(2,-2,IV)
)}}}
When we want to draw an angle on the coordinate plane we put the vertex on the origin, and the side we call the starting side along the positive x-axis.
The other side of the angle is called the terminal side.
{{{drawing(300,300,-5,5,-5,5,
grid(0),
blue(arrow(0,0,4,0)),blue(arrow(0,0,2.8,2.8)),
locate(1,1,45^o),locate(2.8,3.3,terminal),
locate(2.8,2.8,side)
)}}}
 
As solver explains in his answer, the cosine of an angle is the x coordinate of the point where the terminal side crosses the unit circle.
(You could say it is the x coordinate of the point on the terminal side that is one unit away from the origin).
As solver said, the sine of the angle is the y coordinate of that same point.
(By the way, the unit circle solver included in his answer is extremely useful.
If you do not have a printout like that, get one).
 
As for then tangent,
{{{tan(theta)=sin(theta)/cos(theta)}}} , so
{{{tan(theta)=y/x}}} for the point where the terminal side crosses the unit circle.
The beauty of tangent is that you can pick any (x,y) point on the terminal side of the angle to calculate that ratio.
The ratio is the same for all points on the terminal side.
I circled point (1, 1.7) in the drawing below.
{{{drawing(300,300,-5,5,-5,5,
grid(0),circle(1,1.7,0.2),
blue(arrow(0,0,4,0)),blue(arrow(0,0,2.5,4.25)),
locate(0.5,0.6,theta),locate(1.1,1.9,"(1,1.7)")
)}}} {{{tan(theta)=1.7}}} That angle's terminal side is in quadrant I, the first quadrant.
The angle theta in the graph, swept counter-clockwise from starting side to terminal side), is {{{theta=59.5^o}}} (or {{{theta=1.039}}} if measuring in radians).
 
But there are still more answers. The terminal side of an angle with the same tangent could be the ray (on the same line) that starts at the origin, but goes in the opposite direction (see it drawn in green below).
{{{drawing(300,300,-5,5,-5,5,
grid(0),circle(1,1.7,0.2),circle(-1,-1.7,0.2),
blue(arrow(0,0,4,0)),blue(arrow(0,0,2.5,4.25)),
green(arrow(0,0,4,0)),green(arrow(0,0,-2.5,-4.25)),
locate(-3.5,-1.5,"(-1,-1.7)"),locate(1.1,1.9,"(1,1.7)")
)}}} Calling the green angle {{{theta}}}, {{{tan(theta)=(-1.7)/-1=1.7}}} too.
That green angle's terminal side is in quadrant III, the third quadrant.
Now, theta (measured counter-clockwise, as tradition demands) could be {{{180^o+59.5^o}}} (or {{{theta=1.039+pi}}} if using radians).
 
NOTES (more advanced information; skip if confusing): 
We envision the angles as turns, as space swept by a ray (like one of those blue arrows) going from the starting side to the terminal side.
We measure angles as positive if it is a counter-clockwise turn, and negative if it is a clockwise turn.
We even extend the idea of angle to include positive angles, and negative angles whose measure is very large in absolute value , as {{{450^o=1&1/4 turns}}} {{{counter-clockwise}}} , and {{{-760^o=2turns}}} {{{clockwise}}}.
(You may think there is not much use for such large angles, but Shawn White would disagree. Not to mention that a manual could tell you to turn a knob {{{-450^0}}} to mean {{{1&1/4turns}}} {{{clockwise}}}).

The same terminal sides can be reached by going around some number of whole turns (clockwise or counter-clockwise), and then going the {{{59.5^o}}},
or {{{59.5^o+180^o}}} positions counter-clockwise.
All those angles can be represented by
{{{59.5^o+k*180^o}}} for all {{{k}}} integers.
or {{{theta=1.039+k*pi}}} if measuring the angle in radians.