Question 949875: Find sin t and cos t for the given value of t.
t = −13π/4
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! if you use your calculator, you will get:
sin(-13*pi/4) =.7071067812
cos(-13*pi/4) = -.7071057812
your calculator should be in radian mode.
the fact that the answers are the same indicate the reference angle is probably 45 degrees since that only occurs when the reference angle is 45 degrees.
without use of the calculator, you would do the following.
since the angle is negative, keep adding 2*pi to it until it becomes positive.
2*pi is the same as 8*pi/4, so keep adding 8*pi/4 to -13*pi/3 until it becomes positive.
-13*pi/4 + 8*pi/4 = -5*pi/4 + 8*pi/4 = 3*pi/4.
your angle is now positive and is between 0 and 2*pi which is where you want it to be.
an angle of 3*pi/4 is between 2*pi/4 and 4*pi/4.
that puts the angle in the second quadrant.
the reference angle for an angle in the second quadrant is equal to pi - the angle.
pi is equal to 4*pi/4, so your reference angle is 4*pi/4 - 3*pi/4 which is equal to pi/4.
the reference angle is the equivalent angle in the first quadrant.
use your calculator to see that sin(pi/4) = .707... and cos(pi/4) = .707...
convert pi/4 to the equivalent angle in degrees and you can see that pi/4 * 180/pi is equal to 180*pi/4*pi which is equal to 45 degrees.
since you know that sine and cosine of 45 degrees is equal to sqrt(2)/2, then you can convert that to decimal form to see that it is equal to .7071......
it is sometimes easier to convert the angle to degrees from the beginning.
-13*pi/4 * 180/pi is equal to -585 degrees.
keep adding 360 until the angle is between 0 and 360.
-585 + 360 = -225 + 360 = 135.
135 is in the second quadrant.
reference angle is 180 - 135 = 45 degrees.
sine and cosine of 45 degrees is sqrt(2)/2.
that's in the first quadrant.
in the second quadrant, sine is positive and cosine is negative.
sin(135) is therefore equal to sqrt(2)/2.
cos(135) is therefore equal to - sqrt(2)/2.
the sine and cosine will be the same for every complete revolution around the unit circle.
135 - 360 = -225.
sin(-225) = sqrt(2)/2
cos(-225) = - sqrt(2)/2
-225-360 = -585
sin(-585) = sqrt(2)/2
cos(-585) = - sqrt(2)/2
you can use your calculator to confirm.
don't forget to set it to degrees is you're working with degrees.
also sqrt(2)/2 is equal to .7071067812 as indicated before.
your calculator will give you the decimal equivalent.
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