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: $$ (f \cdot g)\left(\frac{\pi}{3}\right) $$

Short answer

\(\frac{\sqrt{3}}{4}\)

Step-by-step solution

- Identify the Functions

First, identify the given functions. They are: - \( f(x) = \sin x \) - \( g(x) = \cos x \)

- Define the Product of Functions

Next, understand that \((f \, \cdot \, g)(x)\) represents the product of the functions \(f(x)\) and \(g(x)\). Thus, \((f \, \cdot \, g)(x) = f(x) \, \cdot \, g(x) = \sin x \, \cdot \, \cos x \).

- Substitute \( \frac{\pi}{3} \) into the Product

Substitute \( x = \frac{\pi}{3} \) into the product obtained in step 2: \[(f \, \cdot \, g)\left(\frac{\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) \, \cdot \, \cos\left(\frac{\pi}{3}\right) \]

- Evaluate the Trigonometric Functions

Recall the values of the trigonometric functions: - \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \) - \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \)

- Calculate the Product

Now calculate the product of the evaluated functions: \[ \sin\left(\frac{\pi}{3}\right) \, \cdot \, \cos\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \, \cdot \, \frac{1}{2} = \frac{\sqrt{3}}{4} \]

Key concepts

sine function

The sine function, written as \( \text{sin}(x) \), is fundamental in trigonometry. It represents the ratio of the length of the side opposite an angle to the hypotenuse in a right-angled triangle. For any angle, \( x \), using the unit circle definition:

\(\text{sin}(x) \) is the y-coordinate of the corresponding point on the unit circle.
Common important angles have values you should memorize:
\(\text{sin}(0) = 0 \)
\(\text{sin}(\frac{\text{π}}{6}) = \frac{1}{2} \)
\(\text{sin}(\frac{\text{π}}{4}) = \frac{\text{√2}}{2} \)
\(\text{sin}(\frac{\text{π}}{3}) = \frac{\text{√3}}{2} \)
\(\text{sin}(\frac{\text{π}}{2}) = 1 \)
It's essential to get comfortable with these values as they are often used in trigonometric problems.

cosine function

The cosine function, denoted as \( \text{cos}(x) \), is another vital trigonometric function. It measures the ratio of the adjacent side to the hypotenuse in a right-angled triangle. For any angle, \( x \), using the unit circle definition:

\(\text{cos}(x) \) is the x-coordinate of the corresponding point on the unit circle.
Key values to remember include:
\(\text{cos}(0) = 1 \)
\(\text{cos}(\frac{\text{π}}{6}) = \frac{\text{√3}}{2} \)
\(\text{cos}(\frac{\text{π}}{4}) = \frac{\text{√2}}{2} \)
\(\text{cos}(\frac{\text{π}}{3}) = \frac{1}{2} \)
These values are just as important as the sine function values, and knowing them helps a lot in problems involving products of trigonometric functions.

product of functions

When dealing with products of functions, like in this exercise where \( f(x) = \text{sin}(x) \) and \( g(x) = \text{cos}(x) \), their product \( (f \cdot g)(x) = \text{sin}(x) \cdot \text{cos}(x) \) is also a function.
Here's what you need to know:

• You multiply the output values of the functions for the same input \( x \). For example, \( f(x) \cdot g(x) = f(2) \cdot g(2) \).
• In trigonometry, this often simplifies using trigonometric identities.
• Evaluate each function separately first before finding their product.

This specific problem involves evaluating \( \text{sin}(\frac{\text{π}}{3}) \) and \( \text{cos}(\frac{\text{π}}{3}) \) before multiplying them.

trigonometric identities

Trigonometric identities are equations involving trigonometric functions that are true for every value in their domain. Here are some essential ones:

The Pythagorean Identity:
\[ \text{sin}^2(x) + \text{cos}^2(x) = 1 \]
It shows the relationship between sine and cosine of the same angle.
Product-to-sum identities can also come in handy:
\[ \text{sin}(x) \cdot \text{cos}(y) = \frac{1}{2} \big[ \text{sin}(x + y) + \text{sin}(x - y) \big] \]
These identities simplify the analysis of the product of trigonometric functions.
In our main example, we used the special values:
\[ \text{sin}\big(\frac{\text{π}}{3}\big) = \frac{\text{√3}}{2} \]
\[ \text{cos}\big(\frac{\text{π}}{3}\big) = \frac{1}{2} \]
Multiplying these gives \(\frac{\text{√3}}{4} \). Trigonometric identities often simplify to these straightforward steps.