Problem 5 - Precalculus with Limits

Question

In Exercises 5-10, verify that the \( x \)-values are solutions of the equation. \( 2 \cos x - 1 = 0 \) (a) \( x = \dfrac{\pi}{3} \) (b) \( x = \dfrac{5\pi}{3} \)

Step-by-Step Solution

Answer

Both \( x = \dfrac{\pi}{3} \) and \( x = \dfrac{5\pi}{3} \) are solutions to the equation \( 2 \cos x - 1 = 0 \).

Step 1: Substitute \( x \) value (a)

Plug \( x = \dfrac{\pi}{3} \) into the equation. The equation \( 2 \cos x - 1 = 0 \) becomes \( 2 \cos \dfrac{\pi}{3} - 1 = 0 \). We know that \( \cos \dfrac{\pi}{3} = \dfrac{1}{2} \). So, the equation becomes \( 2 * \dfrac{1}{2} - 1 = 0 \). That simplifies to \( 1 - 1 = 0 \) which equals to 0. So, \( x = \dfrac{\pi}{3} \) is a solution of the equation.

Step 2: Substitute \( x \) value (b)

Next, substitute \( x = \dfrac{5\pi}{3} \) into the equation. The equation \( 2 \cos x - 1 = 0 \) becomes \( 2 \cos \dfrac{5\pi}{3} - 1 = 0 \). We know that \( \cos \dfrac{5\pi}{3} = \dfrac{1}{2} \). So, the equation becomes \( 2 * \dfrac{1}{2} - 1 = 0 \). That simplifies to \( 1 - 1 = 0 \) which equals to 0. So, \( x = \dfrac{5\pi}{3} \) is also a solution of the equation.

Key Concepts

Cosine Function

The cosine function is an essential part of trigonometry. It relates the angle of a right triangle to the lengths of its adjacent side and hypotenuse. Mathematically, cosine (cos) is defined as the ratio:

• Cosine of angle = Adjacent side / Hypotenuse

The range of the cosine function is between -1 and 1, making it ideal for periodic and wave-like patterns. In the context of trigonometric equations, like the one given in the problem, you can use known values of cosine at specific angles to solve for unknown variables. For example, the cosine of \( \dfrac{\pi}{3} \) and \( \dfrac{5\pi}{3} \) is \( \dfrac{1}{2} \), a fact that simplifies solving equations with these angles.

Solution Verification

Solution verification is a critical step in solving equations. It ensures the values obtained are correct and satisfy the given equation. Let's zoom into the two given values:

• First, for \( x = \dfrac{\pi}{3} \), substitute it into the equation \( 2 \cos x - 1 = 0 \). This becomes \( 2 \times \dfrac{1}{2} - 1 = 0 \). Simplification shows \( 1 - 1 = 0 \) which is correct.
• Secondly, using \( x = \dfrac{5\pi}{3} \), the same substitution yields \( 2 \times \dfrac{1}{2} - 1 = 0 \) leading again to \( 1 - 1 = 0 \), confirming it as a solution.

Through substitution and simplification, you confirm the correctness of solutions.

Unit Circle

The unit circle is a fundamental concept in trigonometry, especially for understanding trigonometric functions. It's a circle with a radius of 1 centered at the origin (0,0) in the coordinate plane. Here's why it's important:

• Every point on the unit circle corresponds to an angle \( x \) and the coordinates of that point are \( (\cos x, \sin x) \).
• This makes calculating cosine and sine values at standard angles straightforward.
• For example, at \( \dfrac{\pi}{3} \) radians, the cosine value, corresponding to the point's x-coordinate, is \( \dfrac{1}{2} \).

The unit circle allows the visualization of these relationships, helping with quicker trigonometric computations.

Radian Measure

Radian measure is a method of measuring angles using the radius of a circle. It is the standard unit of angular measure used in many areas of mathematics. Here's how it works:

• One radian is the angle made at the center of a circle by an arc whose length is equal to the radius.
• A complete circle is \( 2\pi \) radians, equivalent to 360 degrees.

Using radians simplifies many mathematical computations, making it easier to integrate with calculus. In this problem, \( \dfrac{\pi}{3} \) and \( \dfrac{5\pi}{3} \) are radian measures that correspond to angles which are commonly used in trigonometry. They relate directly to positions on the unit circle and the cosine values associated with those angles.