Sin^-1(-√2/2)
To find the inverse sine of a value, we use the notation “sin^-1” or “arcsin”
To find the inverse sine of a value, we use the notation “sin^-1” or “arcsin”. In this case, we want to find the inverse sine of -√2/2.
The inverse sine function returns the angle whose sine is equal to the input value. So, we are looking for an angle whose sine is -√2/2.
First, let’s make sure that the input value, -√2/2, is within the range of the sine function. The sine function has a range of [-1, 1]. Since -√2/2 is between -1 and 1, it is a valid input for the inverse sine function.
To find the angle, we can use the unit circle. The unit circle is a circle with a radius of 1 centered at the origin (0,0) on a coordinate plane. The angle is formed between the positive x-axis and the line connecting the origin and a point on the circle.
For -√2/2, we need to find an angle that has a sine value of -√2/2. This occurs in the third and fourth quadrants of the unit circle. In the third quadrant, the y-coordinate is negative, and in the fourth quadrant, both the x and y-coordinates are positive.
In the third quadrant, the angle is 180 degrees plus the reference angle. The reference angle is the acute angle formed between the x-axis and the line connecting the origin and the point on the unit circle.
Using the Pythagorean theorem, we can find the reference angle. Since the sine of the reference angle is -√2/2, we can say that the y-coordinate is -√2/2 and the x-coordinate is 1 (since it is on the unit circle).
Using the Pythagorean theorem:
1^2 + (-√2/2)^2 = 1 + 2/4 = 1 + 1/2 = 3/2
Since the sum of the squares of the x and y-coordinates is 3/2, we can take the square root to find the length of the hypotenuse which is √(3/2) or √3/√2.
To find the reference angle, we use the fact that sin θ = opposite/hypotenuse. In this case, sin θ = -√2/2 = opposite/√3/√2.
Cross-multiplying, we get:
-√2 * √3 = 2 * opposite
-opposite = √6
So, the length of the opposite side is √6.
Now, we have the reference angle in radians, which can be found using the inverse sine function. The inverse sine of √6/2 gives us the reference angle in radians.
sin^-1(√6/2) ≈ 1.024 radians
Therefore, the angle in radians whose sine is -√2/2 is approximately 1.024 radians.
More Answers:
Understanding the Inverse Sine Function: sin^-1(1/2) Explained with Trigonometry and Reference AnglesThe Complete Guide: Finding the Inverse Sine of 0 and Understanding the Unit Circle
Determining the Angle Whose Sine is -1/2: Finding the Value of sin^-1(-1/2) using the Unit Circle