Question 636737
The Trig ratios are usually taught in terms of opposite, adjacent and hypotenuse. They can also be taught in terms of x and y coordinates and distance from the origin. The correspondence of these is:
x-coordinate = adjacent
y-coordinate = opposite
distance from the origin = hypotenuse<br>
The distance from the origin to some point (x, y) can be expressed using the distance formula. This particular distance is usually called "r" (for reasons you might be able to figure out.). So
{{{r = sqrt((x-0)^2+(y-0)^2) = sqrt(x^2+y^2)}}}
Note: Although x and y coordinates may be negative, r is a distance and it is <i>never</i> negative.<br>
Using the x and y coordiantes given to us in this problem, we can calculate our value for r:
{{{r = sqrt((-3)^2+(4)^2) = sqrt(9+16) = sqrt(25) = 5}}}<br>
We are now ready to find the values of the trig functions. For each function we will replace "adjacent" with the x-coordinate, "opposite" with the y-coordinate, and "hypotenuse" with r we get:
{{{sin(theta) = y/r = 4/5}}}
{{{cos(theta) = x/r = (-3)/5 = -3/5}}}
{{{tan(theta) = y/x = 4/(-3) = -4/3}}}
{{{csc(theta) = r/y = 5/4}}}
{{{sec(theta) = r/x = 5/(-3) = -5/3}}}
{{{cot(theta) = x/y = (-3)/4 = -3/4}}}