Question 1180314
the line is 2x + 7y = 0
solve for y to get:
y = 2/7 * x.


this tells you that:


when the value of x is 0, the value of y is 0.
when the value of x is 1, the value of y is -2/7.


from these 2 value, you can draw a triangle.


the angle at (0,0) is the vertex of the angle.


the tangent of that angle is opposite / adjacent = (-2/7) / 1 = -2/7.


the hypotenuse of that triangle is sqrt(-2/7)^2 + 1^2) = sqrt(53/49).


you have:


adjacent side of angle = 1
opposite side of angle = -2/7
hypotenuse = sqrt(53/49).


let theta represent the angle.


sin(theta) = opposite / hypotenuse = (-2/7) / sqrt(53/49).
cos(theta) = adjacent / hypotenuse = 1 / sqrt(53/49).
tan(theta) = opposite / adjacent = (-2/7) / 1.
cot(theta) - adjacent / opposite = 1 / (-2/7).
sec(theta) = hypotenuse / adjacent = sqrt(53/49) / 1.
csc(theta) = hypotenuse / opposite = sqrt(53/49) / (-2/7).


note that:


cot(theta) = 1 / tan(theta)
sec(theta) = 1 / cos(theta)
csc(theta) = 1 / sin(theta)'


sin = sine
cos = cosine
tan = tangent
cot = cotangent
sec = secant
csc = cosecant


to confirm, you can use your calculator.
arcsin(theta) = (-2/7) / sqrt(53/49) = -15.9453959 degrees.
that's your angle.
let x = -15.9453959 degrees.


sin(x) = -.2747211279
(-2/7) / sqrt(53/49) = the same.


cos(x) = .9615239476.
(1/sqrt(53/49) = the same.


tan(x) = -.2857142857.
(-2/7) / 1 = the same.


cot(x) = 1 / tan(x) = -3.5
1 / (-2/7) = the same.


sec(x) = 1 / cos(x) = 1.040015698.
sqrt(53/49) / 1 the same.


csc(x) = 1 / sin(x) = -3.640054945.
sqrt(53/49) / (-2/7) = the same.


all trig functions point to the same angle.
that angle is -15.9453959 degrees.


that's your angle.


the exact value of the trig functions for that aangle are:


sin(theta) = opposite / hypotenuse = (-2/7) / sqrt(53/49).
cos(theta) = adjacent / hypotenuse = 1 / sqrt(53/49).
tan(theta) = opposite / adjacent = (-2/7) / 1.
cot(theta) - adjacent / opposite = 1 / (-2/7).
sec(theta) = hypotenuse / adjacent = sqrt(53/49) / 1.
csc(theta) = hypotenuse / opposite = sqrt(53/49) / (-2/7).


since the square root of 53/49 is not a rational number, you have to use it as shown and can't simplify it any further and be exact.


sqrt(53/49) = sqrt(53) / sqrt(49).


sqrt(53) is not a rational number.
sqrt(49) is a rational number.


sqrt(53/49) is not a rational number because sqrt(53) is not a rational number.