Find the exact value for sin 315 and csc(22*`pi`/3)

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(a) Since 315 degrees is between 270 degrees and 360 degress, then sin 315 is evaluated in the 4th quadrant, and there is 45 degrees between the terminal arm and the horizontal axis.  In the 4th quadrant, sine is negative, so this means that sin 315 = -sin 45.

But, 45 degrees is the special one - one - root 2 triangle.  So sin 45 = 1 / root 2.  This means that sin 315 = - 1 / root 2, which is approximately  - 0.707

(b) csc pi/3 = 1 / sin pi/3

sin pi/3 is in the first quadrant.  (Remember that pi/3 is 60 degrees.)

pi/3 is part of the special one-two-root 3 triangle, so sin pi/3 = root 3 / 2.

This means that

csc pi/3 = 1 / sin pi/3

= 2 / root 3

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`sin(315^@) = -sin(360^@-315) = -sin(45^@) = -sqrt(2)/2`

`csc((22pi)/3) = 1/sin((22pi)/3)`

`sin((22pi)/3) = sin((4pi)/3 + 6pi) = sin((4pi)/3)) = -sin((4pi)/3 - pi) = -sin(pi/3) = -sqrt(3)/2 `

So

` csc((22pi)/3) = 1/(-sqrt(3)/2) = -(2sqrt(3))/3`

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The exact value of sin 315 and csc (22*`pi` /3) has to be determined.

sin 315 = sin (360 - 45) = -sin 45 = `-1/sqrt 2 ~~ -0.7071`

`csc(22*pi/3) = 1/sin(22*pi/3) = 1/sin(3*2*pi + 4*pi/3)`

=> `1/sin(4*pi/3)`

=> `1/sin(pi + pi/3)`

=> `-1/sin(pi/3)`

=> `-2/sqrt 3`

The value of sin 315 = `-1/sqrt 2` and csc(22*`pi`/3) = `-2/sqrt 3`

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