Question 1180314: Find the exact values of the six trigonometric functions of 𝜃 if 𝜃 is in standard position and the terminal side of 𝜃 is in the specified quadrant and satisfies the given condition.
IV; on the line 2x + 7y = 0
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 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.
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