Sin^-1(-√3/2)
To find the value of arcsin(-√3/2), we need to determine the angle whose sine is -√3/2
To find the value of arcsin(-√3/2), we need to determine the angle whose sine is -√3/2. Recall that the arcsine function, sin^(-1)(x), returns the angle whose sine is x.
First, let’s use the unit circle to understand the concept visually.
The unit circle is a circle with a radius of 1 centered at the origin (0, 0) on a coordinate plane. The unit circle helps us relate the angles in trigonometry with the coordinates on the circle.
The angle whose sine is -√3/2 is in the third and fourth quadrants of the unit circle. In these quadrants, the y-coordinate is negative.
In quadrant III, the values of sin(θ) are negative. So, we can say that sin^(-1)(-√3/2) = θ.
To find θ, we need to determine the reference angle in the first quadrant. The reference angle is the acute angle between the terminal side of the angle and the x-axis.
We know that sin(π/3) = √3/2. Since the reference angle has the same sine value (-√3/2), we can write:
θ = π – (π/3)
θ = 2π/3
So, sin^(-1)(-√3/2) = 2π/3.
Therefore, the value of arcsin(-√3/2) is 2π/3.
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