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: $$ (f \circ h)\left(\frac{\pi}{6}\right) $$

Short answer

The value is \( \frac{\sqrt{3}}{2} \).

Step-by-step solution

- Understand the Composition of Functions

The composition of two functions, say \(f(x)\) and \(g(x)\), written as \((f \circ g)(x)\), is defined as \(f(g(x))\).

- Identify the Inner Function

In this case, the inner function in \((f \circ h)(x)\) is \(h(x)\), which is given as \(h(x) = 2x\). We need to compute \(h\left( \frac{\pi}{6} \right)\).

- Calculate the Inner Function

Substitute \(x = \frac{\pi}{6}\) into the inner function \(h(x)\): \[ h\left( \frac{\pi}{6} \right) = 2 \times \frac{\pi}{6} = \frac{\pi}{3} \]

- Substitute the Result into the Outer Function

Now, take the result from Step 3 and substitute it into the outer function \(f(x)\), which is given as \(f(x) = \sin(x)\). Thus, \[(f \circ h)\left( \frac{\pi}{6} \right) = f\left(\frac{\pi}{3}\right) = \sin\left( \frac{\pi}{3} \right)\]

- Evaluate the Trigonometric Function

Recall that \( \sin\left( \frac{\pi}{3} \right) \) is a standard trigonometric value: \[ \sin\left( \frac{\pi}{3} \right) = \frac{\sqrt{3}}{2} \]

Key concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, especially in areas related to geometry and periodic patterns.
They include sine, cosine, and tangent, among others. These functions help to relate the angles of a triangle to the lengths of its sides.
For example, in the unit circle, the function \(\text{sin}(x)\) gives the y-coordinate of a point on the circle, while \(\text{cos}(x)\) gives the x-coordinate.
The sine function, \(\text{sin}(x)\), oscillates between -1 and 1, which makes it very useful for modeling wave-like phenomena.
Understanding standard trigonometric values, like \(\text{sin}(\frac{\text{π}}{3}) \), is crucial. These values are foundational to many mathematical problems and solutions.

Function Evaluation

Evaluating a function means finding the output of the function for a specific input.
This process involves substituting the given input into the function and performing the necessary calculations.
For example, for a function \(f(x) = \text{sin}(x)\), evaluating \(f(\frac{\text{π}}{3})\) involves substituting \(x \) with \(\frac{\text{π}}{3}\), resulting in \(f(\frac{\text{π}}{3}) = \text{sin}(\frac{\text{π}}{3})\).
It is important to follow the steps critically:

• Identify the input value.
• Substitute it into the function.
• Simplify the expression if needed.

Regular practice with function evaluation enhances understanding and prepares students for more complex problems involving compositions and transformations of functions.

Mathematical Notation

Mathematical notation is the language of mathematics.
It includes symbols, expressions, and formulas used to represent and communicate mathematical ideas.
Composition of functions, denoted as \( (f \circ g)(x) \), signifies applying one function to the result of another. This helps in solving problems systematically.
For example, \( (f \circ h)(\frac{\text{π}}{6}) \) translates to solving \ h(\frac{\text{π}}{6}) \ first, and then applying f to that result.
\( h(x) = 2x \) evaluated at \ \frac{\text{π}}{6} \ gives \ \frac{\text{π}}{3} \. Applying this to f, where \ f(x) = \text{sin}(x) \ results in \ \text{sin}(\frac{\text{π}}{3}) \, which is \ \frac{\text{√3}}{2} \.
Properly understanding and using mathematical notation bridges conceptual gaps and facilitates accurate problem-solving.