how do you find the exact value of sin^-1(-square root of 3 divided by 2)
better just stick with "trig"
One of the first things to do is to learn the special angles with easy trig values: 0,π/6,π/4,π/3,π/2
Knowing those, you will see that
sin π/3 = √3/2
You have -√3/2, so that means that your angle is in
QIII: π+π/3
or
QIV: 2π-π/3
Now, the inverse trig functions have principal values, and for arcsin those are -π/2 <= θ <= π/2
So, Arcsin(-√3/2) = -π/3
To find the exact value of sin^(-1)(-√3/2), also known as sin inverse of (-√3/2), you need to use the unit circle and the properties of the sine function.
1. Start by recalling the definition of the sine function. In trigonometry, sin(x) represents the height of a point on the unit circle at an angle x from the positive x-axis.
2. The unit circle is a circle with a radius of 1 centered at the origin (0, 0) in a coordinate plane. It is helpful to draw a unit circle to visualize the problem.
3. Look for the angle on the unit circle whose sine is -√3/2. Since sine is negative, the angle will be in either the 3rd or 4th quadrant.
4. In the 3rd and 4th quadrants, the sine values are negative. In particular, in the 3rd quadrant, both the x and y values on the unit circle are negative, while in the 4th quadrant, only the y value is negative.
5. Now, consider the special triangles you might remember from trigonometry. One of the special triangles is an equilateral triangle with sides of length 2 and angles of 60 degrees. In this triangle, the sin(60°) = √3/2. Since we are looking for a negative sine value, we need to locate the angle in the 3rd quadrant.
6. The angle in the 3rd quadrant corresponding to -√3/2 is 180° - 60°, which is 120°.
Therefore, sin^(-1)(-√3/2) = 120°.