Problem 25 - Algebra 2 and Trigonometry

Question

If \(\mathrm{f}(x)=\sin \left(\frac{1}{3} x\right),\) find \(\mathrm{f}\left(\frac{\pi}{2}\right)\)

Step-by-Step Solution

Answer

\( f\left(\frac{\pi}{2}\right) = \frac{1}{2} \)

Step 1: Understand the function

The function given is \( f(x) = \sin\left(\frac{1}{3} x\right) \). This function takes an input \( x \), divides it by 3, and then applies the sine function to it.

Step 2: Substitute \( x = \frac{\pi}{2} \) into the function

We need to find the value of \( f\left(\frac{\pi}{2}\right) \), which means substituting \( \frac{\pi}{2} \) for \( x \) in the function definition.Thus, \( f\left(\frac{\pi}{2}\right) = \sin\left(\frac{1}{3} \cdot \frac{\pi}{2}\right) \).

Step 3: Simplify the expression inside the sine function

Let's calculate the expression inside the sine function: \[ \frac{1}{3} \cdot \frac{\pi}{2} = \frac{\pi}{6} \].

Step 4: Compute the sine of \( \frac{\pi}{6} \)

Now compute the sine of the resulting angle: \( \sin\left(\frac{\pi}{6}\right) \).Remember, \( \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \).

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions, often abbreviated as \( \sin \). It is a periodic function, which means it repeats its values in regular intervals. The sine function is defined in the context of a right-angled triangle:

• It describes the ratio of the length of the opposite side to the hypotenuse.
• It varies between -1 and 1 for all angles.

For example, in this exercise, we have the function \( f(x) = \sin\left(\frac{1}{3} x\right) \). This tells us that we take a given angle \( x \), divide it by 3, and find the sine of the resulting angle. Sine waves are an integral part of understanding oscillations and waves, such as sound and light waves.

Function Evaluation

Evaluating a function means finding the function's output for a given input. Think of it like plugging a number into a machine and seeing what comes out. For the function \( f(x) = \sin\left(\frac{1}{3} x\right) \), evaluating \( f\left(\frac{\pi}{2}\right) \) involves:

• Substituting \( \frac{\pi}{2} \) into the function in place of \( x \).
• Simplifying the ensuing expression to get a more straightforward angle.

By substituting \( x = \frac{\pi}{2} \), the expression becomes \( \sin\left(\frac{1}{3} \cdot \frac{\pi}{2}\right) \). Simplifying this expression ensures the calculation is more straightforward, allowing us to find the precise sine value.

Radians

When working with trigonometric functions, you often encounter radians. Radians are a unit of measure for angles, just like degrees. One full circle is \( 2\pi \) radians, equaling 360 degrees. A key reason radians are useful is that they simplify many mathematical calculations, especially those involving calculus.

• \( \pi \) radians is equivalent to 180 degrees.
• The angle \( \frac{\pi}{2} \) radians, used in the exercise, resembles 90 degrees.

In this exercise, \( \frac{\pi}{6} \) radians translates to 30 degrees. Understanding how radians relate to degrees helps visualize the angle's size in familiar terms, which can make it easier to evaluate trigonometric functions correctly.

Angle Simplification

Simplifying the angle within a function can make evaluating a trigonometric function much more manageable. The idea is to reduce the mathematical expression to a simpler form before applying the trigonometric function. Let's break this down:

• When substituting \( \frac{\pi}{2} \) into the function \( f(x) = \sin\left(\frac{1}{3} x\right) \), you compute \( \frac{1}{3} \cdot \frac{\pi}{2} \).
• Perform the multiplication as \( \frac{\pi}{6} \). This angle is much easier to work within trigonometric terms.

Once simplified, the final step is to find the sine of \( \frac{\pi}{6} \), which is \( \frac{1}{2} \). Simplifying the angle beforehand makes the calculation manageable and minimizes the chance of errors.