Find the exact value in degrees sin^-1(-1/square root 2)
the principal value for sin^-1(x) is [-π/2,π/2]
so, since sin^-1(1/√2) = π/4,
sin^-1(-1/√2) = -π/4
You are right. I made an error in my calculations.
The principal value of arcsin(x) is in the range [-π/2, π/2].
Since sin(π/4) = 1/√2, we have arcsin(1/√2) = π/4.
Since sin(-π/4) = -1/√2, we have arcsin(-1/√2) = -π/4.
Therefore, the exact value of sin^-1(-1/√2) is -π/4 or -45 degrees.
The given expression is arcsin(-1/sqrt(2)).
To find the exact value of this expression, we need to determine the angle whose sine is -1/sqrt(2). Remember, arcsin or sin^-1(x) gives the angle that has a sine equal to x.
Since the sine function is negative in the third and fourth quadrants, the angle we are looking for will be in either the third or fourth quadrant.
In the third quadrant, the sine function is negative and its absolute value is 1/sqrt(2) (opposite side divided by hypotenuse in a unit circle). Using the Pythagorean identity (sin^2(theta) + cos^2(theta) = 1), we can determine that the adjacent side in a right triangle formed by this angle is -sqrt(2) - (-1) = -sqrt(2) + 1.
Therefore, in the third quadrant, we have sin(theta) = -1/sqrt(2), cos(theta) = (-sqrt(2) + 1)/sqrt(2) = -sqrt(2)/sqrt(2) + 1/sqrt(2) = -sqrt(2)/2 + 1/sqrt(2) = (-sqrt(2) + 1)/2.
Now, using the inverse tangent function, we can determine the angle in the third quadrant:
tan(theta) = sin(theta)/cos(theta) = (-1/sqrt(2))/((-sqrt(2) + 1)/2) = -2/sqrt(2) = -sqrt(2).
The angle whose tangent is -sqrt(2) in the third quadrant is 225 degrees.
Since sin is a periodic function with a period of 360 degrees, an angle in the fourth quadrant that has the same sine value can be determined by subtracting 360 from the angle in the third quadrant.
In the fourth quadrant, the angle would be 225 - 360 = -135 degrees.
Therefore, the exact value of sin^-1(-1/sqrt(2)) is -135 degrees.