Question

\(f(x)=\sin x, g(x)=\cos x, h(x)=2 x,\) and \(p(x)=\frac{x}{2} .\) Find the value of each of the following: $$ (f \cdot g)\left(\frac{\pi}{4}\right) $$

Short answer

\(\frac{1}{2}\)

Step-by-step solution

Understand the functions

Given functions are: \[ f(x) = \sin x, g(x) = \cos x \]

Define the composite function

We need to find the value of \(f \cdot g\) at x = \frac{\pi}{4}\. The function \(f \cdot g\) is defined as the product of f(x) and g(x): \[ (f \cdot g)(x) = f(x) \cdot g(x) = \sin x \cdot \cos x \]

Evaluate the function at x = \(\frac{\pi}{4}\)

Substitute x = \frac{\pi}{4} in the expression: \[ (f \cdot g)\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) \cdot \cos\left(\frac{\pi}{4}\right)\]

Compute the trigonometric values

We know that \[ \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \text{ and } \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \]

Final calculation

Now, multiply the values: \[ (f \cdot g)\left(\frac{\pi}{4}\right) = \left( \frac{\sqrt{2}}{2} \right) \cdot \left( \frac{\sqrt{2}}{2} \right) = \frac{2}{4} = \frac{1}{2} \]

Key concepts

Product of Functions

In mathematics, when we talk about the product of functions, we refer to multiplying one function by another. For our given problem, we have the functions \(f(x) = \sin x\) and \(g(x) = \cos x\). When we take their product, we create a new function denoted as \((f \cdot g)(x)\). This new function is defined as:
\[(f \cdot g)(x) = f(x) \cdot g(x) = \sin x \cdot \cos x\]
Here, each value of \(x\) will yield a corresponding product of \(\sin x\) and \(\cos x\). This product is sometimes called a pointwise product of the original functions.

Values of Trigonometric Functions

Understanding the values of trigonometric functions at specific angles is crucial. Among the commonly used angles are \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3},\) and \(\frac{\pi}{2}\). Each of these angles corresponds to an easy-to-remember value for both the sine and cosine functions.
For instance, at \(x = \frac{\pi}{4}\):

• \(\sin \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \).
• \(\cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \).

These values are derived from the unit circle and are essential for solving trigonometric problems efficiently.

Evaluating Composite Functions

Evaluating composite functions involves substituting a given value into the composite function. In our example, we need to find \((f \cdot g)\left(\frac{\pi}{4}\right) \). To evaluate this, follow these steps:
1. Identify the composite function: \((f \cdot g)(x) = f(x) \cdot g(x)\).
2. Substitute \(x = \frac{\pi}{4}\) into the composite function:
\( (f \cdot g)\left(\frac{\frac{\pi}{4}}\right) \text{ to get }\sin \(\frac{\pi}{4}\) \cdot \cos\( \frac{\pi}{4}\)\).
3. Use the known values of \( \sin \(\frac{\pi}{4}\)\) and \( \cos\( \frac{\pi}{4}\)\) to get the result.
This approach ensures accuracy and simplifies the process of dealing with composite functions.

Sine Function

The sine function, written as \(\sin x\), is one of the primary trigonometric functions. It is defined as the ratio of the opposite side to the hypotenuse in a right-angled triangle. The function is periodic with a period of \(2\pi\), meaning that \(\sin(x + 2\pi) = \sin x\) for any \(x\).
Some key properties of the sine function are:

• The range is \([-1, 1]\).
• It is symmetric about the origin (odd function): \(\sin(-x) = -\sin(x)\).
• It reaches its maximum value of 1 at \(x = \frac{\pi}{2}\) and minimum value of -1 at \(x = \frac{3\pi}{2}\).

Cosine Function

The cosine function, represented by \(\cos x\), is another fundamental trigonometric function. It is the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Similar to the sine function, the cosine function is periodic with a period of \(2\pi\). This means \(\cos(x + 2\pi) = \cos x\) for any \(x\).
Key properties of the cosine function include:

• The range is \([-1, 1]\).
• It is symmetric about the y-axis (even function): \(\cos(-x) = \cos(x)\).
• It reaches its maximum value of 1 at \(x = 0\) and its minimum value of -1 at \(x = \pi\).

These attributes make the cosine function an essential tool in trigonometric problem-solving.