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Questions and Answers
If sin(θ) = 0.5, what is the value of sin⁻¹(0.5) or asin(0.5)?
What is the range of the arcsine function?
What is the inverse operation of the arcsine function?
If the opposite side of a right triangle is 3 and the hypotenuse is 5, what is the angle (in radians) whose sine corresponds to the ratio 3/5?
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Which of the following statements about the arcsine function is true?
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Study Notes
Arcsine Function
Introduction
The arcsine function, also known as the inverse sine function, is a fundamental concept in trigonometry. It is the inverse operation of the sine function, which maps a real number onto its corresponding angle measure. In practical scenarios, the arcsine function helps to solve problems related to finding the angle whose sine corresponds to a certain ratio of sides in a right triangle. It is commonly represented by the symbols sin⁻¹ or asin and has a domain of [-π/2, π/2].
Definition
Given a ratio of sides in a right triangle, such as opposite side over hypotenuse, the arcsine function can be defined as follows:
If θ = sin⁻¹(opposite side/hypotenuse), where θ is the angle measure, then sin(θ) will equal the given ratio of sides, making it the inverse operation of the sine function. The arcsine function is used to find the value of θ when the ratio of sides is known.
Domain and Range
The domain of the arcsine function is [-π/2, π/2], which means it only applies to angles in this range. The range of the arcsine function is the set of all angles whose sine is equal to the input. For example, if opposite side/hypotenuse = √2/2, then the corresponding angle is 45° (or π/4 radians), which is in the range of the arcsine function.
Uses
The arcsine function is widely used in various fields such as engineering, navigation, physics, and geometry. It helps solve problems where the ratio of sides in a right triangle is known and we need to find the corresponding angle measure. For example, if we know the depth of the ship from the sea bed and the angle the cable makes with the water surface, the arcsine function can help calculate the distance to the bottom of the ship using the sine formula sin(θ) = opposite side/ hypotenuse.
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Description
Explore the concept of the arcsine function, also known as the inverse sine function, in trigonometry. Learn about its definition, domain, range, and practical uses in various fields such as engineering, navigation, physics, and geometry. Understand how the function helps solve problems related to finding angle measures in right triangles.
