Question 1202951: Evaluate the expression:
sin^(-1)(2/5)+sin^(-1)(√21/5)
Found 2 solutions by Theo, greenestamps:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! sin^(-1)(2/5) = 23.57017848 degrees.
sin^*(-1)(sqrt(21)/5) = 66.42182152 degrees.
23.57017848 degrees + 66.42182152 degrees = 90 degrees.
that's your solution.
the angles are complementary.
if you put them in the same right triangle, your triangle would have:
angle A = 23.57017847
angle B = 66.42182152
angle C = 90
given that sin(23.57017848) = 2/5, you have side opposite angle A = 2 and side opposite angle C = 5
given that sin(66.42182152) = sqrt(21) / 5, you have side opposite angle B = sqrt(21) and side opposite angle C = 5.
since side opposite angle A squared plus side opposite angle B squared = side opposite angle C squared, you get (2/5)^2 + (sqrt(21)/5)^2 = 5^2.
this becomes 4/25 + 21/25 = 25 which is true, confirming the value of the angles is correct.
Answer by greenestamps(13203)  (Show Source):
You can put this solution on YOUR website!
Because the inverse sine is positive for both angles, they are both in quadrant I.
Draw a right triangle in standard position with a sine (opposite over hypotenuse) of 2/5 and use  to determine that the cosine of the angle is sqrt(21)/5, which is the sine of the other angle in the given expression.
In a right triangle, the sine of one acute angle is the cosine of the other, so the two angles in the given expression are the two angles in a right triangle. So the sum of the two angles is 90 degrees.
ANSWER: 90 degrees
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