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: $$ (g \circ p)\left(60^{\circ}\right) $$

Short answer

\( (g \circ p)(60^{\text\degree}) = \frac{\sqrt{3}}{2} \)

Step-by-step solution

- Understand the composition of functions

The composition of two functions \((g \circ p)(x)\) means applying the function \((p(x)\) first, and then applying the function \((g(x)\) to the result. Thus, \((g \circ p)(x) = g(p(x))\).

- Apply the function \(p\)

Given \(p(x) = \frac{x}{2}\), apply it to \(60^{\circ}\) to find \(p(60^{\circ})\): \[ p(60^{\circ}) = \frac{60^{\circ}}{2} = 30^{\circ} \]

- Apply the function \(g\) to the result

Next, apply \(g(x) = \cos x\) to the result from Step 2: \[ g(p(60^{\circ})) = g(30^{\circ}) = \cos(30^{\circ}) \]

- Evaluate the cosine function

Use the known value of \(\cos(30^{\circ})\): \[ g(30^{\circ}) = \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \]

Key concepts

Trigonometric Functions

Trigonometric functions are vital in mathematics, especially in dealing with angles and triangles. They help in understanding the relationships between the angles and sides of a right triangle. The primary trigonometric functions are sine (\text{sin}), cosine (\text{cos}), and tangent (\text{tan}).
Trigonometric functions are periodic, which means they repeat their values in regular intervals. For example, the sine and cosine functions have a period of 360 degrees or \(2\text{\textpi}\) radians. This periodicity makes them essential tools for modeling cyclical phenomena like waves. Hence, understanding trigonometric functions is key when dealing with angles and their applications.

Cosine

The cosine function, denoted as \text{cos}(x), is one of the fundamental trigonometric functions. It relates the adjacent side of a right triangle to its hypotenuse. In symbols, for an angle \(x\), the function is defined as:
\text{cos}(x) = \frac{\text{Adjacent}}{\text{Hypotenuse}}
Cosine has several properties:

• It ranges between -1 and 1.
• It has a period of 360 degrees or \(2\text{\textpi}\) radians.
• It is an even function, meaning \text{cos}(-x) = \text{cos}(x).

Also, some key values to remember are \text{cos}(0°) = 1, \text{cos}(90°) = 0, and \text{cos}(30°) = \frac{\text{\textsqrt}{3}}{2}, which are very helpful when solving problems like the one in our exercise.

Function Evaluation

Function evaluation involves finding the output of a function for a given input. When you have a function, like \(f(x) = \textsin(x)\), and you need to find \(f(30°)\), you're essentially asking for the value of the sine function when its input is 30 degrees.
In our exercise, we use the concept of function composition: \(g \textcirc p(60°)\). This means we evaluate \(p\) for 60 degrees first and then use this result to find the value of \(g\). Use these steps:

• Evaluate \(p(60°)\). Given \(p(x) = \frac{x}{2}\), we get \(p(60°) = 30°\).
• Then evaluate \(g(30°)\). Given \(g(x) = \textcos(x)\), we get \(g(30°) = \textcos(30°) = \frac{\textsqrt{3}}{2}\).

This process of step-by-step function evaluation is crucial in breaking down complex problems into manageable parts.

Angle Conversion

Angles can be measured in degrees or radians. Converting between these two is often necessary in trigonometry. The relationship between degrees and radians is given by the following conversion factors:
1 degree = \(\frac{ \textpi}{180}\) radians
1 radian = \(\frac{180}{\textpi}\) degrees
For instance, 30 degrees can be converted to radians as follows:
30° * \(\frac{\textpi}{180}\) = \(\frac{\textpi}{6}\) radians
This conversion is important because trigonometric functions can sometimes require angles in radians. In our exercise, working in degrees simplifies our task but always remember that certain contexts may require radians, particularly in higher mathematics and physics.
Understanding angle conversion aids in better grasping the periodic nature of trigonometric functions and ensures flexibility in various mathematical applications.