Inverse Trigonometric Functions
please explain this to me in mathematical steps please
how to solve sin^(-1) (-1/2)
this equals - pi/6
i know that sin^(-1) domain and range switch from original sin
but i don't know how to apply that... i need an mathematical explanation or if that isn't possible please explain it as simply as possible
thank you in advance
i want to know how it equals -pi/6
To solve sin^(-1) (-1/2), we need to find the angle whose sine is -1/2. This involves using the inverse or "arc" sine function, denoted as sin^(-1) or asin.
Step 1: Recognize the definition of the inverse sine function.
The inverse sine function sin^(-1)(x) or asin(x) gives the angle (in radians) whose sine is x. In other words, sin^(-1) (-1/2) will give us the angle whose sine is -1/2.
Step 2: Determine the reference angle.
To find the reference angle, we need to find the angle in the first quadrant that has the same sine as -1/2. In this case, the reference angle is π/6 (30 degrees) because sin(π/6) = 1/2, and sine is negative in the third quadrant.
Step 3: Apply the domain and range switch.
The domain of the sine function is [-π/2, π/2], which corresponds to the range of the inverse sine function. However, since we are dealing with a negative sine value (-1/2), we need to use the range of the inverse sine function, which is [-π/2, π/2].
Step 4: Determine the complete angle.
Since the sine function is negative in the third quadrant, the complete angle will have the reference angle (π/6) but with a negative sign: -π/6.
Therefore, sin^(-1) (-1/2) = -π/6.
Note: The inverse sine function has multiple values due to the periodic nature of the sine function. In this case, we consider the principal value, which lies within the range [-π/2, π/2].