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Questions and Answers
What is the range of the inverse sine function, sin⁻¹(x)?
What is the range of the inverse sine function, sin⁻¹(x)?
- 0 to π
- 0 to 2π
- −π/2 to π/2 (correct)
- −π to π
Which of the following statements about the function f(x) = sin⁻¹(x) is true?
Which of the following statements about the function f(x) = sin⁻¹(x) is true?
- It produces values outside the range of sine.
- It is only defined for values between -1 and 1. (correct)
- It is defined for all real numbers.
- It is a decreasing function.
What value does sin⁻¹(1/2) represent?
What value does sin⁻¹(1/2) represent?
- π/3
- π/6 (correct)
- 0
- π/4
If y = arcsin(1/3), what is sin(y)?
If y = arcsin(1/3), what is sin(y)?
For x in the domain of the function sin⁻¹(x), which interval can x belong to?
For x in the domain of the function sin⁻¹(x), which interval can x belong to?
What is the value of tan(arcsin(1/3))?
What is the value of tan(arcsin(1/3))?
What condition must x satisfy for sin⁻¹(x) to exist?
What condition must x satisfy for sin⁻¹(x) to exist?
Which of these is NOT a property of the sine function?
Which of these is NOT a property of the sine function?
Why do trigonometric functions need their domains restricted?
Why do trigonometric functions need their domains restricted?
What is the primary role of the Horizontal Line Test in relation to functions?
What is the primary role of the Horizontal Line Test in relation to functions?
Which of the following is the correct notation for the inverse sine function?
Which of the following is the correct notation for the inverse sine function?
Which of the following intervals makes the sine function one-to-one?
Which of the following intervals makes the sine function one-to-one?
What is the inverse function of f(x) = sin(x) when restricted appropriately?
What is the inverse function of f(x) = sin(x) when restricted appropriately?
What is the effect of restricting the sine function's domain on its graph?
What is the effect of restricting the sine function's domain on its graph?
How can the existence of an inverse function be defined mathematically?
How can the existence of an inverse function be defined mathematically?
In inverse trigonometric functions, what is arcsin(y) asking for?
In inverse trigonometric functions, what is arcsin(y) asking for?
What is the value of $sin(3)$ when calculated in radian mode?
What is the value of $sin(3)$ when calculated in radian mode?
Which of the following represents the correct value of $sec(3)$?
Which of the following represents the correct value of $sec(3)$?
If $ heta$ is in degree mode, how is $sin(3 ext{°})$ expressed?
If $ heta$ is in degree mode, how is $sin(3 ext{°})$ expressed?
What is the reciprocal of $sin(3)$ known as?
What is the reciprocal of $sin(3)$ known as?
What is the value of $cot(3)$?
What is the value of $cot(3)$?
In radians, what is the expression for $csc(3)$?
In radians, what is the expression for $csc(3)$?
What is the assumed mode of the calculator when calculating $sin(3)$?
What is the assumed mode of the calculator when calculating $sin(3)$?
If the angle is given in degrees, what formula should be used to find $tan(3)$?
If the angle is given in degrees, what formula should be used to find $tan(3)$?
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Study Notes
Inverse Sine Function (arcsin)
- The inverse sine function, denoted by sin−1 or arcsin, is defined as the inverse of the restricted sine function.
- The restricted sine function is defined as f(x) = sin(x), where -π/2 ≤ x ≤ π/2.
- This restricted function is one-to-one, meaning that for every y-value there is only one corresponding x-value.
- The inverse sine function satisfies the following equation: sin−1(x) = y ⇔ sin(y) = x, where -π/2 ≤ y ≤ π/2 and -1 ≤ x ≤ 1.
- In other words, sin−1(x) is the angle between -π/2 and π/2 whose sine is x.
Evaluating Inverse Sine Function (arcsin)
- To evaluate sin−1(x), you find the angle between -π/2 and π/2 whose sine is x.
- For example, sin−1(1/2) = π/6 because sin(π/6) = 1/2 and π/6 lies within the range -π/2 to π/2.
Calculating with Inverse Sine Function (arcsin)
- To calculate tan(arcsin(1/3)), follow these steps:
- Let θ = arcsin(1/3).
- This means sin(θ) = 1/3.
- Visualize a right triangle with opposite side 1 and hypotenuse 3 (because sin(θ) = opposite/hypotenuse).
- Use the Pythagorean theorem to find the adjacent side: √(3^2 - 1^2) = √8 = 2√2.
- Now, tan(θ) = opposite/adjacent = 1/(2√2) = √2/4.
- Therefore, tan(arcsin(1/3)) = √2/4.
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