Arcsine

Definition

The arcsine function, also known as the inverse sine function, is a trigonometric function that allows you to find the angle whose sine is a given value. It is the inverse operation of the sine function, providing the angle measurement in radians given the sine value as input.

5 Must Know Facts For Your Next Test

1. The arcsine function is denoted as $\sin^{-1}(x)$ or $\arcsin(x)$, where $x$ is the input value.
2. The arcsine function is used to solve trigonometric equations where the sine function is involved.
3. The arcsine function is particularly useful when solving for an unknown angle in a right triangle, given the value of the sine ratio.
4. The arcsine function is an odd function, meaning $\sin^{-1}(-x) = -\sin^{-1}(x)$.
5. The arcsine function is a one-to-one function, meaning each input value has a unique output value within the defined domain and range.

Review Questions

• Explain how the arcsine function is used to solve trigonometric equations involving the sine function.
  • The arcsine function is used to solve trigonometric equations where the sine function is involved. By applying the arcsine function to both sides of the equation, the unknown angle can be isolated and expressed in radians. For example, if the equation is $\sin(\theta) = 0.5$, applying the arcsine function to both sides gives $\theta = \sin^{-1}(0.5) = \pi/6$ radians. This allows the angle $\theta$ to be determined from the given sine value.
• Describe the domain and range of the arcsine function and explain their significance in solving trigonometric equations.
  • The domain of the arcsine function is the interval $[-1, 1]$, and the range is the interval $[-\pi/2, \pi/2]$ in radians. The significance of these boundaries is that the arcsine function is only defined for input values between -1 and 1, as these are the valid values for the sine function. Any input value outside this range will result in an undefined output. Additionally, the range of $[-\pi/2, \pi/2]$ radians represents all possible angle measurements that have a sine value within the valid domain. This range constraint ensures that the arcsine function provides a unique angle solution for each valid input value.
• Analyze the relationship between the arcsine function and the sine function, and explain how this relationship can be used to verify solutions to trigonometric equations.
  • The arcsine function is the inverse of the sine function, meaning that if $y = \sin(x)$, then $x = \sin^{-1}(y)$. This inverse relationship can be used to verify solutions to trigonometric equations involving the sine function. For example, if the solution to the equation $\sin(\theta) = 0.5$ is $\theta = \pi/6$ radians, then applying the arcsine function to both sides should give $\sin^{-1}(\sin(\pi/6)) = \pi/6$, confirming the validity of the solution. This verification process is crucial in ensuring that the solutions obtained using the arcsine function are correct and satisfy the original trigonometric equation.

Related terms

Inverse Trigonometric Functions: Inverse trigonometric functions, such as arcsine, arccosine, and arctangent, are used to find the angle given the value of a trigonometric ratio.

Radians: Radians are a unit of angle measurement, where one radian is the angle subtended by an arc equal in length to the radius of the circle.

Domain and Range: The domain of the arcsine function is the interval $[-1, 1]$, and the range is the interval $[-\pi/2, \pi/2]$ in radians.