Problem 20 - Algebra 2

Question

Find each value. Write angle measures in radians. Round to the nearest hundredth. $$ \sin \left(\sin ^{-1} \frac{1}{2}\right) $$

Step-by-Step Solution

Answer

The value is \( \frac{1}{2} \).

Step 1: Understand the Problem

We need to determine the value of the expression \( \sin \left( \sin^{-1} \frac{1}{2} \right) \). This involves finding the sine of the angle whose sine value is \( \frac{1}{2} \).

Step 2: Solve Inverse Sine Function

The inverse sine function, \( \sin^{-1} \), gives the angle whose sine is a given number. Here, we seek the angle whose sine is \( \frac{1}{2} \). In the range from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), \( \sin^{-1} \frac{1}{2} = \frac{\pi}{6} \).

Step 3: Evaluate the Sine of the Angle

Now that we have determined that \( \sin^{-1} \frac{1}{2} = \frac{\pi}{6} \), we need to find \( \sin \left( \frac{\pi}{6} \right) \). From trigonometric values, \( \sin \left( \frac{\pi}{6} \right) = \frac{1}{2} \).

Step 4: Final Verification of Result

Since \( \sin \left( \frac{\pi}{6} \right) = \frac{1}{2} \), our solution confirms \( \sin \left( \sin^{-1} \frac{1}{2} \right) = \frac{1}{2} \). The solution is consistent and verified.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are a fascinating aspect of mathematics that allow us to find angles given trigonometric values. Specifically, the inverse sine function, denoted as \( \sin^{-1}(x) \), helps determine the angle whose sine is \( x \). Unlike regular trigonometric functions that take angles as inputs and provide values, inverse functions reverse this process.
For example, when we calculate \( \sin^{-1}(\frac{1}{2}) \), we are looking for the angle \( \theta \) such that \( \sin(\theta) = \frac{1}{2} \).
Inverse trigonometric functions have specific ranges to ensure they are functions (with one output for each input).

• The range for \( \sin^{-1}(x) \) is from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).

Using these functions allows us to transition smoothly from trigonometric values to angles, which is immensely useful when solving real-world problems or verifying mathematical identities.

Radians

Radians are a way of measuring angles that connects geometry with the properties of circles more naturally than degrees do. One radian is the angle created when the radius of a circle is wrapped along its circumference. This unit of measure provides a direct connection to the circle itself.
The formula used to convert degrees to radians is \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). This allows one to seamlessly work with standard angle measures and trigonometric functions without constant conversions.
Radians are especially useful in calculus and other advanced mathematical fields, as they simplify mathematical expressions and make derivatives and integrals involving trigonometric functions more straightforward. For instance:

• Using radians, \( \sin\left(\frac{\pi}{2}\right) = 1 \) is a simple and direct calculation.

Using radians is a natural approach to solving problems involving periodic phenomena such as waves and oscillations because the radian measure ties directly to the length of the arc on a unit circle.

Sine Function

The sine function is one of the most fundamental trigonometric functions, which relates the angle in a right triangle to the ratio of the length of the opposite side to the hypotenuse. Its values cycle between -1 and 1 as the angle varies from \( 0 \) to \( 2\pi \) radians.
In this context, understanding \( \sin(x) \) involves recognizing the cyclical, smooth wave it creates, known as a sinusoidal wave.
Key properties of the sine function include:

• It is periodic with a period of \( 2\pi \), meaning \( \sin(x) = \sin(x + 2\pi n) \) for any integer \( n \).
• Its maximum value is 1, minimum is -1, and it crosses zero at integer multiples of \( \pi \).

In solving the exercise, knowing \( \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \) reflects these properties, showing how the function maps an angle to its corresponding value. The sine function is essential not only in mathematics but also in modeling waves, sound, light, and many other scientific phenomena.