Problem 96 - Algebra and Trigonometry

Question

let $$ f(x)=\sin x, g(x)=\cos x, \text { and } h(x)=2 x $$ Find the exact value of each expression. Do not use a calculator. $$ (h \cdot f)\left(\frac{11 \pi}{4}\right) $$

Step-by-Step Solution

Answer

The exact value of \((h \cdot f)(\frac{11 \pi}{4})\) is \(\frac{11 \pi \sqrt{2}}{4}\).

Step 1: Substituting \(x\) in \(f(x)\)

First, put \(\frac{11 \pi}{4}\) into \(f\), we get \(f(\frac{11 \pi}{4}) = \sin(\frac{11 \pi}{4})\). Using the periodic property of sine function, \(\sin(x + 2n\pi) = \sin(x)\), where \(n\) is an integer, we simplify this as \(\sin(\frac{3 \pi}{4})\).

Step 2: Calculating \(\sin(\frac{3 \pi}{4})\)

The value of \(\sin(\frac{3 \pi}{4})\) is \(\frac{\sqrt{2}}{2}\), based on the unit circle definition.

Step 3: Substituting \(x\) into \(h(x)\)

Now, let's substitute \(\frac{11 \pi}{4}\) into \(h(x)\). Therefore, \(h(\frac{11 \pi}{4}) = 2 * \frac{11 \pi}{4}\), which simplifies to \(\frac{11 \pi}{2}\).

Step 4: The multiplication of functions \((h \cdot f)(x)\)

The multiplication of functions means multiplying the outputs of functions \(f\) and \(h\). Therefore, we multiply \(\frac{\sqrt{2}}{2}\) (found in Step 2) with \(\frac{11 \pi}{2}\) (found in Step 3). The result is \(\frac{11 \pi \sqrt{2}}{4}\).

Key Concepts

Understanding the Sine Function

The sine function is a fundamental trigonometric function often denoted as \( \sin x \). It is used to determine the vertical component of a point on the unit circle at a particular angle \( x \). This function takes any real number as input and maps it onto the interval \([-1, 1]\).

• It is defined as the ratio of the opposite side to the hypotenuse in a right triangle.
• For unit circle coordinates, \( (\sin x, \cos x) \) represent the y-coordinate of the point at angle \( x \).
• The sine function is periodic with a period of \(2\pi\), meaning that \( \sin(x) = \sin(x + 2n\pi) \) for any integer \( n \).

Understanding how sine works with the unit circle helps simplify problems like \( \sin(\frac{11\pi}{4}) \), using its periodic nature.

Decoding the Cosine Function

The cosine function, denoted as \( \cos x \), is another critical trigonometric function that is closely related to the sine function. It provides the horizontal component of an angle on the unit circle. Like sine, it also maps real numbers onto the interval \([-1, 1]\).

• The cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse.
• In the unit circle, \( \cos x \) gives the x-coordinate of the point corresponding to an angle \( x \).
• Similar to the sine function, cosine is periodic with a period of \(2\pi\), such that \( \cos(x) = \cos(x + 2n\pi) \).

Being familiar with both sine and cosine functions enhances understanding of their behavior over one cycle of the unit circle.

Exploring Function Multiplication

Function multiplication involves creating a new function that is the product of two given functions.

• Given two functions \( f(x) \) and \( h(x) \), the product function \((f \cdot h)(x)\) is defined as \( f(x) \times h(x) \).
• This is also known as pointwise multiplication, where values at each \( x \) from the respective functions are multiplied together.

Applying this concept to \((h \cdot f)(\frac{11\pi}{4})\), we substitute the values of \( f \) and \( h \) obtained at \( \frac{11\pi}{4} \), then multiply them to find the result, as shown in the solution.

Navigating the Unit Circle

The unit circle is a fundamental tool in trigonometry that provides insights into the values of trigonometric functions at various angles. It is a circle with a radius of one, centered at the origin of a coordinate plane.

• Each angle \( x \) in radians on the unit circle corresponds to a point \((\cos x, \sin x)\).
• Angles are typically measured from the positive x-axis, moving counterclockwise for positive angles and clockwise for negative angles.
• The unit circle is useful for quickly determining standard angle values for trigonometric functions.

In the solution, to evaluate \( \sin(\frac{11\pi}{4}) \), the periodic nature aids in reducing \( \sin \) to \( \sin(\frac{3\pi}{4}) \), leveraging the unit circle's reference angles.

Grasping Periodic Properties

Periodic properties of trigonometric functions make them valuable for a wide range of mathematical applications. These properties allow the functions to repeat their values at regular intervals.

• Sine and cosine functions both have the same period of \(2\pi\), meaning their graphs repeat every \(2\pi\) units.
• This periodicity implies that for any \( x \), \( \sin(x + 2n\pi) = \sin(x) \) and \( \cos(x + 2n\pi) = \cos(x) \), where \( n \) is an integer.
• Understanding periodic properties is crucial for simplifying expressions and solving trigonometric equations.

In the original problem, periodic properties of sine simplify \( \sin(\frac{11\pi}{4}) \) to \( \sin(\frac{3\pi}{4}) \), a more manageable expression.