Inverse Sine Function (arcsin) Overview

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Questions and Answers

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?

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?

π/3

π/6 (correct)

π/4

If y = arcsin(1/3), what is sin(y)?

1/3 Signup and view all the answers

For x in the domain of the function sin⁻¹(x), which interval can x belong to?

−1 to 1 Signup and view all the answers

What is the value of tan(arcsin(1/3))?

√2/3 Signup and view all the answers

What condition must x satisfy for sin⁻¹(x) to exist?

−1 ≤ x ≤ 1 Signup and view all the answers

Which of these is NOT a property of the sine function?

It is an even function. Signup and view all the answers

Why do trigonometric functions need their domains restricted?

To make them one-to-one and enable inverse functions Signup and view all the answers

What is the primary role of the Horizontal Line Test in relation to functions?

To check if a function has an inverse by verifying one-to-one behavior Signup and view all the answers

Which of the following is the correct notation for the inverse sine function?

Both A and B are correct Signup and view all the answers

Which of the following intervals makes the sine function one-to-one?

−$rac{ ext{π}}{2}$ to $rac{ ext{π}}{2}$ Signup and view all the answers

What is the inverse function of f(x) = sin(x) when restricted appropriately?

sin^-1(x) Signup and view all the answers

What is the effect of restricting the sine function's domain on its graph?

The graph can pass the Horizontal Line Test Signup and view all the answers

How can the existence of an inverse function be defined mathematically?

f(f^-1(x)) = x for all x in the domain Signup and view all the answers

In inverse trigonometric functions, what is arcsin(y) asking for?

The angle whose sine is y Signup and view all the answers

What is the value of $sin(3)$ when calculated in radian mode?

0.14112 Signup and view all the answers

Which of the following represents the correct value of $sec(3)$?

$-2$ Signup and view all the answers

If $ heta$ is in degree mode, how is $sin(3 ext{°})$ expressed?

$sin(3 ext{°})$ Signup and view all the answers

What is the reciprocal of $sin(3)$ known as?

cosecant Signup and view all the answers

What is the value of $cot(3)$?

$-rac{1}{ ext{s}3}$ Signup and view all the answers

In radians, what is the expression for $csc(3)$?

$rac{ ext{c}3}{ ext{s}3}$ Signup and view all the answers

What is the assumed mode of the calculator when calculating $sin(3)$?

Radian mode Signup and view all the answers

If the angle is given in degrees, what formula should be used to find $tan(3)$?

$ ext{tan}(3 ext{°})$ Signup and view all the answers

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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Description

This quiz covers the inverse sine function, commonly known as arcsin, detailing its definition, restrictions, and evaluation methods. Learn how to calculate with arcsin and understand its geometric interpretation through right triangles. Test your knowledge of the properties and applications of this fundamental trigonometric concept.