Trigonometry is a branch of mathematics that deals with the relationship between angles and the sides of a right-angled triangle. The sine function is a fundamental trigonometric function that relates the ratio of the length of the side opposite an angle to the length of the hypotenuse. The inverse sine function, also known as arcsine, is the mathematical operation that undoes the sine function. It allows us to find the measure of an angle when given the ratio of the opposite side to the hypotenuse.
What is Sine?
Sine is one of the six basic trigonometric functions and is defined as the ratio of the length of the side opposite an angle to the length of the hypotenuse in a right-angled triangle. In mathematical terms, sine is expressed as:
sin θ = Opposite side / Hypotenuse
The value of sine ranges from -1 to 1, depending on the angle θ. For example, the sine of a 90-degree angle is equal to 1, while the sine of a 180-degree angle is equal to 0.
What is Inverse Sine?
The inverse sine function, denoted as sin-1 or arcsin, is the mathematical operation that undoes the sine function. It allows us to find the measure of an angle when given the ratio of the opposite side to the hypotenuse. In other words, if sin θ = x, then sin-1 x = θ.
It is important to note that the inverse sine function is not the same as raising sine to the power of -1. The notation sin-1 x represents the inverse sine function, while (sin x)-1 represents the reciprocal of sine, which is cosecant.
Inverse Sine Formula
The inverse sine function is used to find the measure of an angle when given the ratio of the opposite side to the hypotenuse. The formula for inverse sine is as follows:
θ = sin-1 (Opposite side / Hypotenuse)
This formula allows us to determine the angle θ when given the value of the ratio of the opposite side to the hypotenuse.
Graph of Inverse Sine Function
The graph of the inverse sine function, also known as the arcsine graph, represents the values of the inverse sine for different inputs. The domain of the inverse sine function is -1 to 1, inclusive, and the range is -π/2 to π/2, inclusive.
The graph of the inverse sine function is a curve that starts at (-1, -π/2) and (1, π/2), representing the minimum and maximum values for the inverse sine function. The curve is symmetric with respect to the y-axis, reflecting the symmetry of the sine function. As the input approaches -1 or 1, the output of the inverse sine function approaches -π/2 or π/2, respectively.
Inverse Sine Derivative
The derivative of the inverse sine function, denoted as d/dx sin-1 x, can be calculated using the chain rule of differentiation. The derivative of the inverse sine function with respect to x is given by:
d/dx sin-1 x = 1 / √(1 - x^2)
This formula allows us to find the rate of change of the inverse sine function with respect to x at any given point.
Inverse Sine Table
The values of the inverse sine function for different ratios of the opposite side to the hypotenuse can be represented in a table. The table below shows the values of sin-1 x for different values of x:
| x | sin-1 x (in radians) | sin-1 x (in degrees) |
|---|---|---|
| -1 | -π/2 | -90° |
| -√3/2 | -π/3 | -60° |
| -√2/2 | -π/4 | -45° |
| -1/2 | -π/6 | -30° |
| 0 | 0 | 0° |
| 1/2 | π/6 | 30° |
| √2/2 | π/4 | 45° |
| √3/2 | π/3 | 60° |
| 1 | π/2 | 90° |
These values represent the angles for which the sine function returns the given ratio of the opposite side to the hypotenuse.
Proof of Derivative of Sin Inverse x
The derivative of the inverse sine function, d/dx sin-1 x, can be derived using the chain rule of differentiation. Let y = sin-1 x. By the definition of the inverse sine function, sin y = x. Differentiating both sides with respect to x, we get:
cos y (dy/dx) = 1
dy/dx = 1 / cos y
Using the Pythagorean identity sin^2 y + cos^2 y = 1, we can express cos y as √(1 – sin^2 y) = √(1 – x^2). Substituting this value, we have:
dy/dx = 1 / √(1 - x^2)
This formula allows us to find the derivative of the inverse sine function with respect to x at any given point.
Anti-Derivative of Sin Inverse x
The anti-derivative of the inverse sine function, also known as the integral of sin-1 x, can be determined using integration by parts method. Let’s consider the integral of sin-1 x with respect to x:
∫ sin-1 x dx
By using integration by parts, assuming u = sin-1 x and dv = dx, we can write the formula as:
∫ u dv = uv - ∫ v du
Integrating both sides, we have:
∫ sin-1 x dx = x sin-1 x + ∫ x / √(1 - x^2) dx
Using the u-substitution method, where u = 1 – x^2, du = -2x dx, we can further simplify the integral:
∫ x / √(1 - x^2) dx = -∫ du / 2√u
Now, integrating ∫ du / 2√u, we get:
-∫ du / 2√u = -√u = -√(1 - x^2)
Substituting this value back into the original equation, we have:
∫ sin-1 x dx = x sin-1 x - √(1 - x^2) + C
This formula allows us to find the anti-derivative of the inverse sine function with respect to x.
Properties of Inverse Sine Function
The inverse sine function, or arcsin, has several properties that are useful in solving trigonometric problems. These properties are as follows:
- sin(sin-1 x) = x, for -1 ≤ x ≤ 1
- sin-1(sin x) = x, for -π/2 ≤ x ≤ π/2
- sin-1(-x) = -sin-1 x, for -1 ≤ x ≤ 1
- sin-1(1/x) = csc-1 x, for |x| ≥ 1
- sin-1 x + cos-1 x = π/2, for -1 ≤ x ≤ 1
These properties allow us to simplify trigonometric expressions and solve equations involving the inverse sine function.
Domain and Range of Inverse Sine
The domain of the inverse sine function, sin-1 x, is the set of real numbers from -1 to 1, inclusive. This means that the input value x must be between -1 and 1 in order for the inverse sine function to be defined.
The range of the inverse sine function is the set of real numbers from -π/2 to π/2, inclusive. This means that the output value of the inverse sine function, θ, must lie between -π/2 and π/2 in order to satisfy the definition of the inverse sine function.
How to Find Sin Inverse x?
To find the inverse sine of x, follow these steps:
- Recognize that the possible answers lie in the range of -π/2 to π/2.
- Assume that y is the inverse sine of x, denoted as y = sin-1 x. According to the definition of the inverse sine function, this means sin y = x.
- Consider values of y within the interval of -π/2 to π/2 that satisfy the equation sin y = x. The solution lies within this range.
For example, to find sin-1(0.5):
- Recognize that the possible answers lie in the range of -π/2 to π/2.
- Assume that y is the inverse sine of 0.5, denoted as y = sin-1(0.5). According to the definition of the inverse sine function, this means sin y = 0.5.
- Consider values of y within the interval of -π/2 to π/2 that satisfy the equation sin y = 0.5. The solution lies within this range.
In this case, y = sin-1(0.5) = π/6 or 30°.
Derivative and Integral of Inverse Sine
The derivative of the inverse sine function, denoted as d/dx sin-1 x, is given by:
d/dx sin-1 x = 1 / √(1 - x^2)
This formula allows us to find the rate of change of the inverse sine function with respect to x at any given point.
The anti-derivative of the inverse sine function, also known as the integral of sin-1 x, can be determined using integration by parts method. The integral of sin-1 x with respect to x is given by:
∫ sin-1 x dx = x sin-1 x - √(1 - x^2) + C
This formula allows us to find the anti-derivative of the inverse sine function with respect to x.
Solved Examples of Inverse Sine
- Find sin-1(1/2).
Solution: We know that sin-1(1/2) = π/6 or 30°.
- Find the derivative of sin-1 x.
Solution: The derivative of sin-1 x is given by d/dx sin-1 x = 1 / √(1 – x^2).
- Evaluate ∫ sin-1 x dx.
Solution: The integral of sin-1 x with respect to x is given by ∫ sin-1 x dx = x sin-1 x – √(1 – x^2) + C.
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