this means the ratio of opposite over adjacent is equal to 11.
this means that, regardless of the length of the adjacent side, the opposite side always has to be 11 times that.
you can use the inverse trig function to find the angle.
the angle is equal to arctan(11) = 84.80557109 degrees = 84.806 degrees rounded to 3 decimal places.
once you have the angle, you can find the other trig functions.
if we let theta = 84.80557109 degrees, then:
sin(theta) = .9958932065 =.996 rounded to 3 decimal places.
cos(theta) = .090535746= .091 rounded to 3 decimal places.
tan(theta) = 11
cot(theta) = 1/tan(theta) = .0909090909..... =.091 rounded to 3 decimal places.
sec(theta) = 1/cos(theta) = 11.04536102 = 11.045 rounded to 3 decimal places.
csc(theta) = 1/sin(theta) = 1.004123729 = 1.004 rounded to 3 decimal places.
graphically it would look like this:
on the graph, the blue line is the value of the indicated trig function and the black line is the value of the angle that is the arctan(11).
the intersection of the blue line with the black line is the indicated trig function of that angle.
for example:
sin(x) = .996 when x = 84.806
84.806 degrees is the arctan of 11.
.996 is the sine of 84.806 degrees after it's been rounded to 3 decimal places.
here's the graphs:
you could also get the same results without finding the angle.
if tan(theta) = 11, then the side opposite = 11 and the side adjacent the angle = 1
find the hypotenuse by use of pythagorus to get c^2 = 1^2 + 11^1 = 122.
this makes c (the hypotenuse) = sqrt(122) = 11.04536102
sin(theta) = opp/hyp = 11/11.04536102 = .9958932065
likewise:
cos(theta) = adj/hyp = 1/11.04536102 = .090535746
tan(theta) = opp/adj = ....
cot(theta) = 1/tan(theta) = ....
sec(theta) = 1/cos(theta) = ....
csc(theta) = 1/sin(theta) = ....
you can do the math.
you will find that the value of the trig functions are the same as we got by finding the angle first and then finding the value of the trig function of that angle.
in the graph, x is used instead of theta.
it's just a name change.
the angle is the same.