Question 1153127: What is the exact value of arccos(sin(-4/7)) without using a calculator?
I drew out the triangle of sin(-4/7) with it being in the 3rd quadrant, and found the other side to be -sqrt(33). I am unsure of what to do next. Thank you so much for your help!
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13203)   (Show Source):
You can put this solution on YOUR website!
The expression you show that is to be evaluated is almost certainly not right....
In the expressions sin(x) or cos(x), x has to be the measurement of an angle; and the measurement is in radians unless degrees is specified.
In the expression arcsin(x) or arccos(x), x has to be the value of the sine or cosine.
So the expression sin(-4/7) which is part of the expression as you show it means the sine of an angle of -4/7 radians. But it is highly probable that the -4/7 is supposed to be the sine of the angle (ratio of opposite side to hypotenuse, as in the calculation you did).
In that case the correct expression to be evaluated is probably cos(arcsin(-4/7)).
Note that arcsin(-4/7) is an angle in the 4th quadrant, where cosine is positive.
Then, using your right triangle in the 4th quadrant, the opposite side is -4 and the hypotenuse is 7, making the adjacent side sqrt(33), as you calculated.
Then the cosine of that angle is adjacent/hypotenuse = sqrt(33)/7.
Answer by ikleyn(52814)  (Show Source):
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