Problem 97 - 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. the average rate of change of \(f\) from \(x_{1}=\frac{5 \pi}{4}\) to \(x_{2}=\frac{3 \pi}{2}\)

Step-by-Step Solution

Answer

The exact value of the average rate of change of f() from \(x_{1}=\frac{5 \pi}{4}\) to \(x_{2}=\frac{3 \pi}{2}\) is \( -2\sqrt{2}/\pi\)

Step 1: Compute f(x_1) and f(x_2)

Start by evaluating \(f(x)=\sin x\) at \(x_{1}=\frac{5 \pi}{4}\) and \(x_{2}=\frac{3 \pi}{2}\). Recall that the sine of \(5\pi/4\) and \(3\pi/2\) are \(-\frac{\sqrt{2}}{2}\) and \(-1\) respectively. Therefore we have \(f(x_{1}) =-\frac{\sqrt {2}}{2}\) and \(f(x_{2})=-1\)

Step 2: Compute the average rate of change

Apply these values to the formula for average rate of change (f(x_2) - f(x_1))/(x_2 - x_1). Substitute \(f(x_{1}) =-\frac{\sqrt {2}}{2}\), \(f(x_{2})=-1\), \(x_{1}=\frac {5 \pi}{4}\), \(x_{2}=\frac {3 \pi}{2}\) into the formula. This yields to \(((-1) - (-\frac{\sqrt {2}}{2}))/((\frac{3 \pi}{2})-(\frac{5 \pi}{4}))\) which simplifies to \((-\frac{\sqrt {2}}{2})/(\frac{\pi}{4})\).

Step 3: Simplify the rate of change

Finally, carry out the division to simplify the average rate of change expression to \( -2\sqrt{2}/\pi\)

Key Concepts

Trigonometric Functions

Trigonometric functions are crucial in mathematics and physics. They describe relationships between the angles and sides of triangles, going beyond right-angled triangles to apply broadly in wave functions and circular motion. These functions are periodic, meaning they repeat their values in regular intervals.

The main trigonometric functions are sine, cosine, and tangent, often represented by the symbols sin, cos, and tan, respectively. Each has a specific relationship to the unit circle, a circle with a radius of one centered at the origin of a graph.

• The sine function considers the vertical component, or the y-value, of a point on the unit circle.
• The cosine function relates to the horizontal component, or the x-value, of the same point.
• The tangent function involves the ratio of the sine to the cosine for a given angle.

In our original exercise, we used sine and cosine functions to calculate the average rate of change. Understanding how these functions interact with angles, such as those measured in radians, is key to solving problems in trigonometry.

It’s important to remember that trigonometric functions can model periodic phenomena such as sound waves and tides, making them invaluable across multiple scientific disciplines.

Sine Function

The sine function, denoted as \( \sin x \), is one of the fundamental functions in trigonometry. It outputs the y-coordinate of a point on the unit circle as the angle, \( x \), varies. The range of the sine function is between -1 and 1, reflecting the height of the point on the unit circle at different angles.

In our example, we calculated the sine of angles \( \frac{5\pi}{4} \) and \( \frac{3\pi}{2} \), which represent specific positions on the unit circle:

• For \( x = \frac{5\pi}{4} \), the sine function has a value of \(-\frac{\sqrt{2}}{2}\). This occurs in the third quadrant of the unit circle.
• For \( x = \frac{3\pi}{2} \), the sine function reaches \(-1\), corresponding to the bottommost point on the unit circle.

The sine function is continuous and smooth, repeating its values every \( 2\pi \) radians. This periodic behavior makes it vital for modeling oscillating systems like acoustics and alternating electric currents.

Learning about the sine function's graph can help visualize and predict its behavior in mathematical models or real-world phenomena.

Pi as a Constant

Pi, denoted as \( \pi \), is an essential constant in mathematics, especially concerning geometry and trigonometry. Pi is approximately 3.14159 and represents the ratio of a circle's circumference to its diameter. This ratio is consistent for all circles, making \( \pi \) a universal constant.

In the context of our exercise, \( \pi \) is used to express angles in radians. Radians are a measure of angle that relates directly to the arc length of the unit circle, where one complete revolution is \( 2\pi \) radians.

Using radians, trigonometric functions like \( \sin x \) and \( \cos x \) become particularly elegant and predictable.

• The angle \( \frac{5\pi}{4} \) corresponds to a point well into the third quadrant of the unit circle.
• The angle \( \frac{3\pi}{2} \) reaches the nadir, or lowest point, consistent with negative sine values.

Understanding \( \pi \) and its impact on trigonometric calculations ensures accuracy in both academic and applied mathematics, spanning fields from engineering to physics and beyond.