Question

find the exact value without using a calculator. $$ \sin \left(\sin ^{-1} 0.4567\right) $$

Short answer

\( \sin(\sin^{-1}(0.4567)) = 0.4567 \).

Step-by-step solution

Understand the Inverse Function

The function \( \sin^{-1}(x) \) is the inverse of the sine function, which means it's used to find the angle \( \theta \) whose sine value is \( x \). In this exercise, \( x = 0.4567 \). We want to determine the angle \( \theta \) such that \( \sin(\theta) = 0.4567 \).

Apply the Definitions of Sine and Inverse Sine

Since \( \sin^{-1}(0.4567) \) gives us the angle whose sine is 0.4567, by definition, \( \sin(\sin^{-1}(0.4567)) = 0.4567 \). Therefore, \( \theta = \sin^{-1}(0.4567) \) and when you take the sine of \( \theta \), you get back 0.4567.

Key concepts

Sine Function

The sine function is a basic trigonometric function that relates to the angle in a right triangle to the ratio of the length of the opposite side to the hypotenuse. It is defined as:
\[\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}\]This function is widely used in mathematics, physics, and engineering because it can model wave-like phenomena such as sound and light waves. Understanding the sine function allows us to explore properties of triangles and various periodic phenomena.
Some key features of the sine function include:

• The range of values for sine is between -1 and 1, inclusive.
• The sine function is periodic with a period of \(2\pi\), meaning it repeats every \(2\pi\) radians.
• The sine function is continuous and smooth for all values of \(\theta\).

These characteristics make the sine function fundamental in trigonometry.

Inverse Functions

Inverse functions are a significant concept in mathematics, allowing us to reverse the effect of a function. For the sine function, the inverse is the arcsine function, denoted by \(\sin^{-1}(x)\). It provides the angle whose sine is a given number.
When you see \(\sin^{-1}(x)\), it refers to the angle \(\theta\) such that \(\sin(\theta) = x\). In essence, if \(\sin(\theta) = x\), then \(\theta = \sin^{-1}(x)\).
Key points to remember about inverse functions include:

• The domain of \(\sin^{-1}(x)\) is \([-1, 1]\), since these are the only valid outputs for the sine of an angle.
• The range of \(\sin^{-1}(x)\) is \([-\frac{\pi}{2}, \frac{\pi}{2}]\), covering angles from -90 to 90 degrees.
• Using inverse functions, we can solve equations involving trigonometric functions for angles.

Understanding inverse functions, like the arcsine, is essential for handling trigonometric equations and analyzing phenomena with wave-like patterns.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. They are fundamental tools in simplifying and solving trigonometric equations.
One commonly used identity is the reciprocal identity:

• \(\sin(\sin^{-1}(x)) = x\)

This identity tells us that if you take the sine of an arcsine, you return to your original value \(x\), so long as \(x\) is within the radius where trigonometric functions maintain their usual range (in this case, \([-1, 1])\).
Other important trigonometric identities include:

• Pythagorean identities, such as \(\sin^2(\theta) + \cos^2(\theta) = 1\).
• Angle sum and difference identities, which allow calculation of trigonometric functions for sums and differences of angles.
• Double angle identities, which relate the trigonometric functions of twice an angle to the functions of the angle.

Mastering trigonometric identities allows for deeper insights and simplifications when working with complex trigonometric problems.