Exact Value Of Arcsin(√2/2) A Step-by-Step Guide

In the realm of mathematics, particularly within trigonometry and inverse trigonometric functions, finding the exact value of expressions is a fundamental skill. This article delves into the process of determining the precise value of the inverse sine function, specifically arcsin(√2/2). We will explore the underlying concepts, step-by-step methodology, and the significance of this value within the broader context of mathematics.

The arcsin function, denoted as sin⁻¹(x) or arcsin(x), is the inverse of the sine function. It answers the question: "What angle has a sine of x?" The domain of the arcsin function is [-1, 1], and its range is [-π/2, π/2] radians or [-90°, 90°] degrees. This range restriction is crucial because the sine function is periodic, meaning it repeats its values over intervals. To define a unique inverse, we limit the output of arcsin to this specific range.

To find the exact value of arcsin(√2/2), we need to identify the angle within the range of [-π/2, π/2] whose sine is √2/2. Recall the unit circle, a circle with a radius of 1 centered at the origin of a coordinate plane. Points on the unit circle are defined by their coordinates (cos θ, sin θ), where θ is the angle formed between the positive x-axis and the line segment connecting the origin to the point.

Consider the special right triangle with angles 45°-45°-90°. This triangle has sides in the ratio 1:1:√2. When this triangle is placed within the unit circle, with the right angle at the origin and one leg along the positive x-axis, the angle formed at the origin is 45° or π/4 radians. The coordinates of the point where the hypotenuse intersects the unit circle are (√2/2, √2/2). Therefore, sin(π/4) = √2/2.

Since π/4 falls within the range of the arcsin function [-π/2, π/2], we can conclude that arcsin(√2/2) = π/4 radians or 45 degrees. This is the exact value of the expression.

Understanding Inverse Trigonometric Functions

To truly grasp the concept of finding the exact value of arcsin(√2/2), it is crucial to have a solid understanding of inverse trigonometric functions. These functions, also known as arcus functions, are the inverses of the basic trigonometric functions: sine, cosine, and tangent. They play a vital role in various fields, including mathematics, physics, engineering, and computer science. Inverse trigonometric functions allow us to determine the angle corresponding to a given trigonometric ratio.

The Concept of Inverse Functions

Before delving into inverse trigonometric functions, let's briefly review the concept of inverse functions in general. An inverse function essentially "undoes" the operation of the original function. If we have a function f(x) that maps a value x to a value y, then the inverse function, denoted as f⁻¹(y), maps y back to x. In other words, f⁻¹(f(x)) = x and f(f⁻¹(y)) = y.

For a function to have a true inverse, it must be one-to-one, meaning that each input value maps to a unique output value. However, trigonometric functions are periodic, which means they repeat their values over intervals. As a result, they are not one-to-one over their entire domains. To define inverse trigonometric functions, we need to restrict the domains of the original trigonometric functions to intervals where they are one-to-one.

Restricting the Domains of Trigonometric Functions

To define the inverse sine function (arcsin), we restrict the domain of the sine function to the interval [-π/2, π/2]. Over this interval, the sine function is one-to-one, and its range is [-1, 1]. Similarly, for the inverse cosine function (arccos), we restrict the domain of the cosine function to the interval [0, π]. Over this interval, the cosine function is one-to-one, and its range is [-1, 1]. For the inverse tangent function (arctan), we restrict the domain of the tangent function to the interval (-π/2, π/2). Over this interval, the tangent function is one-to-one, and its range is (-∞, ∞).

The Arcsin Function

The arcsin function, denoted as sin⁻¹(x) or arcsin(x), is the inverse of the sine function. It takes a value x in the interval [-1, 1] as input and returns an angle θ in the interval [-π/2, π/2] such that sin(θ) = x. In other words, arcsin(x) answers the question: "What angle has a sine of x?"

Understanding the Range of Arcsin

The range of the arcsin function is [-π/2, π/2], which corresponds to the first and fourth quadrants of the unit circle. This restriction is crucial because the sine function is positive in both the first and second quadrants. To define a unique inverse, we limit the output of arcsin to the first and fourth quadrants. When finding the arcsin of a positive value, the result will be an angle in the first quadrant. When finding the arcsin of a negative value, the result will be an angle in the fourth quadrant (expressed as a negative angle).

Applying Arcsin to Solve Problems

Understanding the arcsin function is essential for solving various problems in mathematics and other fields. For instance, it is used to find angles in triangles, solve trigonometric equations, and analyze periodic phenomena. By knowing the range and properties of arcsin, we can accurately determine the angles that satisfy given conditions.

The Unit Circle and Special Angles

The unit circle is a powerful tool for understanding trigonometric functions and their inverses. It is a circle with a radius of 1 centered at the origin of a coordinate plane. Points on the unit circle are defined by their coordinates (cos θ, sin θ), where θ is the angle formed between the positive x-axis and the line segment connecting the origin to the point.

The unit circle provides a visual representation of the values of sine, cosine, and tangent for different angles. By understanding the unit circle, we can easily determine the values of these trigonometric functions for special angles, such as 0°, 30°, 45°, 60°, and 90°, as well as their multiples and negative counterparts.

Special Angles and Their Trigonometric Values

Certain angles, known as special angles, have trigonometric values that can be expressed exactly using radicals and fractions. These angles are particularly important in trigonometry and related fields. The special angles commonly encountered are:

• 0° (0 radians)
• 30° (π/6 radians)
• 45° (π/4 radians)
• 60° (π/3 radians)
• 90° (π/2 radians)

The trigonometric values for these special angles are summarized in the following table:

Using the Unit Circle to Find Trigonometric Values

The unit circle allows us to quickly determine the sine, cosine, and tangent values for any angle. To find the sine of an angle, we look at the y-coordinate of the point where the angle intersects the unit circle. To find the cosine of an angle, we look at the x-coordinate of the point. To find the tangent of an angle, we divide the y-coordinate by the x-coordinate.

For example, to find sin(45°), we locate the point on the unit circle corresponding to 45° (π/4 radians). The coordinates of this point are (√2/2, √2/2). Therefore, sin(45°) = √2/2. Similarly, cos(45°) = √2/2 and tan(45°) = 1.

Finding Arcsin Values Using the Unit Circle

The unit circle is also helpful in finding the values of inverse trigonometric functions, such as arcsin. To find arcsin(x), we look for the angle on the unit circle whose sine (y-coordinate) is equal to x. We then identify the angle within the range of arcsin [-π/2, π/2] that corresponds to this point.

In the case of arcsin(√2/2), we look for the point on the unit circle where the y-coordinate is √2/2. This point corresponds to an angle of 45° (π/4 radians). Since 45° falls within the range of arcsin, we conclude that arcsin(√2/2) = π/4 radians.

Quadrantal Angles

Quadrantal angles are angles that lie on the axes of the coordinate plane. These angles are 0°, 90°, 180°, 270°, and 360° (or 0, π/2, π, 3π/2, and 2π radians). The trigonometric values for quadrantal angles are particularly easy to determine using the unit circle.

• 0° (0 radians): The point on the unit circle is (1, 0). sin(0°) = 0, cos(0°) = 1, tan(0°) = 0.
• 90° (π/2 radians): The point on the unit circle is (0, 1). sin(90°) = 1, cos(90°) = 0, tan(90°) is undefined.
• 180° (π radians): The point on the unit circle is (-1, 0). sin(180°) = 0, cos(180°) = -1, tan(180°) = 0.
• 270° (3π/2 radians): The point on the unit circle is (0, -1). sin(270°) = -1, cos(270°) = 0, tan(270°) is undefined.
• 360° (2π radians): The point on the unit circle is (1, 0). sin(360°) = 0, cos(360°) = 1, tan(360°) = 0.

Step-by-Step Solution for arcsin(√2/2)

Now, let's revisit the original problem and provide a detailed, step-by-step solution for finding the exact value of arcsin(√2/2).

1. Understand the arcsin function: Recall that arcsin(x) is the inverse of the sine function. It returns the angle whose sine is x, within the range of [-π/2, π/2] radians or [-90°, 90°] degrees.

2. Identify the sine value: We are given arcsin(√2/2), so we need to find the angle whose sine is √2/2.

3. Recall special angles: From our knowledge of special angles and the unit circle, we know that sin(45°) = sin(π/4) = √2/2.

4. Check the range: The angle 45° (π/4 radians) falls within the range of arcsin [-π/2, π/2].

5. State the solution: Therefore, arcsin(√2/2) = π/4 radians or 45 degrees.

Significance and Applications

The ability to find the exact values of inverse trigonometric functions, such as arcsin(√2/2), is crucial for several reasons:

• Mathematical Precision: Exact values are essential for accurate calculations and proofs in mathematics. Approximations, while useful in some contexts, can lead to errors in more complex calculations.

• Understanding Trigonometric Relationships: Finding exact values helps solidify our understanding of the relationships between angles and trigonometric ratios. It allows us to visualize these relationships on the unit circle and connect them to geometric concepts.

• Applications in Physics and Engineering: Trigonometric functions and their inverses are widely used in physics and engineering to model periodic phenomena, solve problems involving triangles, and analyze waves and oscillations. Exact values are often required for precise calculations in these fields.

• Computer Graphics and Game Development: Trigonometric functions and their inverses are fundamental in computer graphics and game development for tasks such as rotations, transformations, and lighting calculations. Exact values can improve the accuracy and realism of these applications.

Conclusion

In conclusion, finding the exact value of arcsin(√2/2) is a fundamental exercise in trigonometry that reinforces our understanding of inverse trigonometric functions, the unit circle, and special angles. By following a step-by-step approach and utilizing our knowledge of these concepts, we can confidently determine that arcsin(√2/2) = π/4 radians or 45 degrees. This skill is essential for various applications in mathematics, physics, engineering, and computer science.

The mastery of finding exact values of trigonometric and inverse trigonometric functions lays a strong foundation for more advanced mathematical concepts and problem-solving. By understanding the underlying principles and practicing these techniques, we can develop a deeper appreciation for the beauty and power of mathematics.

A. arcsin(√2/2) = π/4