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csc 240 = hypotenuse divided by opposite = h/o.
sin 240 = opposite divided by hypotenuse = o/h
csc 240 = 1 / sin 240 = 1 / (o/h) = h/o
using your calculator:
sin 240 = -.866024404
csc 240 degrees = 1 / sin 240 = 1 / -.866024404 = -1.154700538
the sine of 240 is negative because the angle is in the 3d quadrant.
in the 3d quadrant, the x-value is negative and the y-value is also negative.
the x-value is negative because it is to the left of the y-axis.
the y-value is negative because it is below the x-axis.
the hypotenuse, or the line segment forming the angle, is always positive.
that's why the sin 240 is negative.
since the csc 240 = 1 / sin 240, the csc 240 is also negative.
the angle of 240 is equal to 180 + 60 degrees.
the trigonometric functions of an angle of 240 degrees is equivalent to the trigonometric functions of an angle of 60 degrees except for the sign.
sin (60) = .866025404
csc (60) = 1/.866025404 = 1.154700538
csc (60) and sin (60) are the same as csc(240) and sin(240) except for the sign.
the sine of 60 degrees is positive since the angle of 60 degrees is in the 1st quadrant.
in the 1st quadrant, the x-value and the y-value are both positive.
sine of the angle = opposite divided by hypotenuse = positive value divided by positive value = positive.
if you have a scientific calculator that calculates sine, cosine, tangent, then you don't really need to know anything other than that the cosecant equals 1 divided by the sine of an angle.
this is because most scientific calculators only have sine, cosine, and tangent functions, and do not have cosecant, secant, and cotangent functions.
the relationships are:
cosecant of an angle equals 1 divided by the sine of the angle.
secant of an angle equals 1 divided by the cosine of the angle.
cotangent of an angle equals 1 divided by the tangent of the angle.
all you needed to do was take the sine of 240 degrees and then divide that into 1.
the calculator takes care of everything else, including the sign of the result.
to really understand what is actually going on, you need to know a little bit more about angles and their trigonometric functions and the signs of trigonometric functions in the different quadrants of the unit circle, as well as the acute angle equivalents of obtuse angles.
one tutorial that describes such things can be found at the following link.
there are other, some of which are referenced in the linked tutorial.
you can also view lessons written by other tutors on algebra.com by going to the following link and looking for all lessons on trigonometry.