Problem 61
Question
Use the properties of inverse functions to find the exact value of the expression, if possible. \(\sin ^{-1}\left(\sin \frac{5 \pi}{2}\right)\)
Step-by-Step Solution
Verified Answer
The exact value of the expression is \(\frac{\pi}{2}\)
1Step 1: Identify the original angle
The original angle in the expression provided is \(\frac{5 \pi}{2}\)
2Step 2: Adjust original angle
Given the sine function has a periodicity of \(2\pi\), adjust the provided angle such that it falls within the interval \([- \pi, \pi]\). The adjusted angle is \(\frac{5 \pi}{2} - 2\pi = \frac{\pi}{2}\)
3Step 3: Apply properties of inverse functions
Apply the principle that inverse sine of sine of an angle equals the angle itself, which is only true if the angle is in the range \([- \frac{\pi}{2}, \frac{\pi}{2}]\). Here, the adjusted angle \(\frac{\pi}{2}\) is at the boundary of the specified interval. So, \(\sin ^{-1}\left(\sin \frac{\pi}{2}\right) = \frac{\pi}{2}\)
Key Concepts
Sine FunctionPeriodicity of Trigonometric FunctionsInverse Trigonometric Properties
Sine Function
Trigonometric functions are fundamental in mathematics, describing the relationships between the sides and angles of triangles. The sine function is one of these trigonometric functions and is particularly significant because of its wide-ranging applications in science, engineering, and mathematics.
Specifically, the sine of an angle in a right-angled triangle is defined as the ratio of the length of the opposite side to the hypotenuse. Symbolically, this is expressed as \( \sin(\theta) = \frac{opposite}{hypotenuse} \). The sine function takes an angle as input and produces a real number between -1 and 1 as its output. It's worth noting that this function is defined not just for acute angles in triangles, but for any real-number angle, being applicable in unit circles and wave functions.
Understanding how to manipulate and interpret the sine function is crucial because it lays the foundation for more advanced topics in trigonometry and calculus, such as integration and Fourier analysis. It also aids in solving problems involving periodic phenomena, like sound waves and light waves.
Specifically, the sine of an angle in a right-angled triangle is defined as the ratio of the length of the opposite side to the hypotenuse. Symbolically, this is expressed as \( \sin(\theta) = \frac{opposite}{hypotenuse} \). The sine function takes an angle as input and produces a real number between -1 and 1 as its output. It's worth noting that this function is defined not just for acute angles in triangles, but for any real-number angle, being applicable in unit circles and wave functions.
Understanding how to manipulate and interpret the sine function is crucial because it lays the foundation for more advanced topics in trigonometry and calculus, such as integration and Fourier analysis. It also aids in solving problems involving periodic phenomena, like sound waves and light waves.
Periodicity of Trigonometric Functions
Trigonometric functions are periodic, meaning they repeat their values in regular intervals. The period of a function is the distance over which the function's graph completes one full pattern before starting to repeat itself. The sine function, in particular, has a period of \(2\pi\) radians, which means that after every \(2\pi\) interval, the sine values repeat.
An important consequence of periodicity is that when working with angles for these functions, you can often add or subtract multiples of \(2\pi\) to find an equivalent angle that yields the same function value. This is useful in simplifying calculations and is crucial when working with inverse trigonometric functions that are restricted to specific domains.
As periodicity can also be interpreted in terms of motion along the unit circle, it serves as the foundational concept in describing oscillatory motion such as that of pendulums, vibrations, and alternating currents.
An important consequence of periodicity is that when working with angles for these functions, you can often add or subtract multiples of \(2\pi\) to find an equivalent angle that yields the same function value. This is useful in simplifying calculations and is crucial when working with inverse trigonometric functions that are restricted to specific domains.
As periodicity can also be interpreted in terms of motion along the unit circle, it serves as the foundational concept in describing oscillatory motion such as that of pendulums, vibrations, and alternating currents.
Inverse Trigonometric Properties
Inverse trigonometric functions are the reverse processes of the trigonometric functions. They take a real number as input, typically representing the ratio between sides of a triangle or a point on the unit circle, and output an angle—the original angle that would produce the given sine, cosine, etc.
These inverse functions have specific properties that are crucial for solving trigonometric equations. One of these properties is the restricted domain. For instance, the inverse sine function, denoted \(\sin^{-1}\) or arcsine, is only defined for inputs between -1 and 1 and within a range of \[ -\frac{\pi}{2}, \frac{\pi}{2} \]. The reason for this restriction is to provide a unique angle for every possible value.
Applying these properties correctly is imperative when trying to simplify expressions involving inverse trigonometric functions. For example, when faced with \(\sin^{-1}(\sin(\theta))\), if \(\theta\) falls within the restricted domain, you can simplify directly to \(\theta\). If not, adjustments to \(\theta\) must be made using the concept of periodicity to bring it within the allowed range before applying the inverse function.
These inverse functions have specific properties that are crucial for solving trigonometric equations. One of these properties is the restricted domain. For instance, the inverse sine function, denoted \(\sin^{-1}\) or arcsine, is only defined for inputs between -1 and 1 and within a range of \[ -\frac{\pi}{2}, \frac{\pi}{2} \]. The reason for this restriction is to provide a unique angle for every possible value.
Applying these properties correctly is imperative when trying to simplify expressions involving inverse trigonometric functions. For example, when faced with \(\sin^{-1}(\sin(\theta))\), if \(\theta\) falls within the restricted domain, you can simplify directly to \(\theta\). If not, adjustments to \(\theta\) must be made using the concept of periodicity to bring it within the allowed range before applying the inverse function.
Other exercises in this chapter
Problem 61
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