Problem 79 - Precalculus with Limits

Question

In Exercises 77-80, find all solutions of the equation in the interval \( [0, 2\pi) \). Use a graphing utility to graph the equation and verify the solutions. \( \cos \dfrac{x}{2} - \sin x = 0 \)

Step-by-Step Solution

Answer

The solutions to the equation \( \cos \dfrac{x}{2} - \sin x = 0 \) in the interval [0, 2\pi) are \( x = \pi \) and \( x = \pi/3 \).

Step 1: Isolate cosine term

Rearrange the equation to isolate the cosine term on one side. So the equation becomes \( \cos \dfrac{x}{2} = \sin x \)

Step 2: Use the double-angle formula

Recall that \( \sin x = \cos (\dfrac{\pi}{2} - x) \). So, we have \( \cos \dfrac{x}{2} = \cos (\dfrac{\pi}{2} - x) \).

Step 3: Find solutions using the cosine function property

Since the cosine function has the property \( \cos a = \cos b \), either \( a = b \) or \( a = -b \). Setting those equalities gives the possible solutions: \( x/2 = \pi/2 - x \) or \( x/2 = -\pi/2 + x \). These can be solved to give \( x = \pi \) or \( x = \pi/3 \).

Step 4: Verify solutions by graph

Graph the equation \( y = \cos \dfrac{x}{2} - \sin x \) along with the equation \( y = 0 \). The intersection points are the solutions, and they should correspond to \( x = \pi \) and \( x = \pi/3 \).

Key Concepts

Cosine Function

The cosine function is a fundamental concept in trigonometry. It's one of the primary trigonometric functions, often abbreviated as "cos." The cosine of an angle in a right-angled triangle is the ratio of the adjacent side to the hypotenuse. There are several important features of the cosine function:

• Periodicity: The cosine function is periodic with a period of \(2\pi\). This means after \(2\pi\), the function values repeat.
• Range: The range of the cosine function is from -1 to 1. It's maximum value is 1, and its minimum value is -1.
• Symmetry: Cosine is an even function, which means \(\cos(-x) = \cos(x)\).

Understanding these properties is essential when solving trigonometric equations involving cosine, such as in the given equation \( \cos \dfrac{x}{2} = \sin x \). Knowing that cosine and sine functions are related through transformations can help simplify and solve complex equations.

Sine Function

The sine function is another primary trigonometric function, abbreviated as "sin." It is defined for an angle in a right triangle as the ratio of the opposite side to the hypotenuse. Important aspects of the sine function include:

• Periodicity: Like the cosine function, sine is also periodic with a period of \(2\pi\).
• Range: The range for the sine function is from -1 to 1, just like cosine.
• Symmetry: The sine function is an odd function, which results in \(\sin(-x) = -\sin(x)\).
• Phase Shift: Sine and cosine differ by a phase shift of \(\pi/2\) radians. This is why \(\sin x = \cos(\pi/2 - x)\).

This relationship between sine and cosine is vital in solving equations where both functions are present. In the equation \( \cos \dfrac{x}{2} = \sin x \), recognizing that sine can be expressed in terms of cosine allows us to equate the terms for further simplification.

Intersection Points

Finding intersection points in trigonometric equations involves determining where two graphs meet on the coordinate plane. In the problem \( \cos \dfrac{x}{2} - \sin x = 0 \), the goal is to find values of \(x\) where the two functions are equivalent. Here's how you can do it:

• Graph the individual functions: Plot \( y = \cos \dfrac{x}{2} \) and \( y = \sin x \) separately.
• Identify intersection points: Look for where the graphs intersect. These points are solutions to the original equation.

To verify solutions, often a graphing utility is used. The intersection points should confirm the solutions we found analytically, like \(x = \pi/3\) and \(x = \pi\). By picturing where the graphs meet, you can visually see where both expressions equal each other. This graphical approach complements the algebraic method and solidifies understanding.

Cosine and Sine Properties

The properties of cosine and sine are crucial for solving equations that involve these trigonometric functions. They provide useful identities and transformations that simplify complex expressions:

• Cosine and Sine Transformation: Since \( \sin x = \cos(\pi/2 - x) \), you can transform sine into cosine, which helps when functions are mixed.
• Complementary Angles: Cosine and sine of complementary angles add to 90 degrees, leading to equal values, i.e., \( \cos A = \sin(90^\circ - A) \).
• Identities: The Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) is one of the most commonly used. It's helpful in converting between sine and cosine.

Understanding these properties allows us to solve trigonometric equations more effectively by converting them into simpler forms, like turning \( \cos \dfrac{x}{2} - \sin x = 0 \) into separate cosine terms, which can be solved using properties like phase shifts and periodicity.