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: $$ (p \circ g)\left(45^{\circ}\right) $$

Short answer

\(\frac{\sqrt{2}}{4}\)

Step-by-step solution

Identify given functions

The given functions are: \(f(x)=\sin x\), \(g(x)=\cos x\), \(h(x)=2 x\), and \(p(x)=\frac{x}{2}\).

Understand the composition of functions

The composition \((p \circ g)(x)\) means applying \(g(x)\) first and then applying \(p(x)\) to the result of \(g(x)\). Mathematically, this is written as \(p(g(x))\).

Calculate \(g(45^{\circ})\)

We need to find the value of \(g(45^{\circ})\). Since \(g(x) = \cos x\), we have: \[ g(45^{\circ}) = \cos 45^{\circ} \]. From trigonometry, we know: \[ \cos 45^{\circ} = \frac{\sqrt{2}}{2} \].

Apply the function \(p(x)\)

Now, use the result of \(g(45^{\circ})\) in the function \(p(x) = \frac{x}{2}\). So, we need to find \(p\left(\frac{\sqrt{2}}{2}\right)\).

Calculate \(p\left(\frac{\sqrt{2}}{2}\right)\)

Substitute \(\frac{\sqrt{2}}{2}\) into the function \(p(x)\): \[ p\left(\frac{\sqrt{2}}{2}\right) = \frac{\frac{\sqrt{2}}{2}}{2} = \frac{\sqrt{2}}{4} \].

Key concepts

Trigonometric Functions

Trigonometric functions help us understand the relationships between the angles and sides of a right triangle. These functions are essential in various fields, such as physics and engineering, where wave patterns, oscillations, and rotations are analyzed. In this exercise, we specifically use the sine and cosine functions.

The cosine function, denoted as \(\cos x\), measures the ratio of the adjacent side to the hypotenuse in a right triangle. For example, \(\cos 45^{\circ}\) is \(\frac{\sqrt{2}}{2}\). This value is derived from the properties of an isosceles right triangle, where the two non-hypotenuse sides are equal, leading the hypotenuse to be \(\sqrt{2}\) times longer than each leg.

Function Composition

Function composition is a way to combine two functions to form a new function. This technique allows us to apply one function to the results of another function.

For example, if we have functions \(f(x)\) and \(g(x)\), the composition \(f(g(x))\) means we first apply \(g(x)\) to some input \(x\), then take that result and apply \(f\) to it. In our exercise, we deal with \(g(x) = \cos x\) and \(p(x) = \frac{x}{2}\), and we want to find the value of \( (p \circ g)(45^{\circ})\), which is \( p(g(45^{\circ})) \).

We first calculate \( g(45^{\circ}) = \cos 45^{\circ} = \frac{\sqrt{2}}{2} \), and then apply the function \( p \) to this result: \( p\left( \frac{\sqrt{2}}{2}\right) = \frac{\frac{\sqrt{2}}{2}}{2} = \frac{\sqrt{2}}{4} \).

Evaluating Functions

Evaluating functions means finding the output of a function for a specific input. This requires us to substitute the given input into the function's formula.

Let's break it down step-by-step using the exercise example. We need to evaluate the composition of functions \( (p \circ g)(45^{\circ}) \).

Start by identifying the given input and the functions: the composition means we use \( g(45^{\circ})\) first. Since \( g(x) = \cos x \), we evaluate \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \).

Next, use \( \cos 45^{\circ}\)'s result as the input for the next function \( p(x) = \frac{x}{2} \). We substitute \( \frac{\sqrt{2}}{2} \) into function \( p \)'s formula to get: \( p\left( \frac{\sqrt{2}}{2} \right) = \frac{\frac{\sqrt{2}}{2}}{2} = \frac{\sqrt{2}}{4} \).

Thus, the value of \( (p \circ g)(45^{\circ}) \) is \( \frac{\sqrt{2}}{4} \). Breaking down these steps helps in understanding the detailed process of function evaluation.