Problem 43 - Precalculus with Limits

Question

In Exercises 39-44, solve the multiple-angle equation. \( \cos \dfrac{x}{2} = \dfrac{\sqrt{2}}{2} \)

Step-by-Step Solution

Answer

The solutions of the equation \( \cos \dfrac{x}{2} = \dfrac{\sqrt{2}}{2} \) are \(x = \pi/2\) or \(x = 7\pi/2\)

Step 1: Identifying Quadrants

The cosine function could have a positive value such as \(\sqrt{2}/2\) in two quadrants, namely the first and the fourth quadrants.

Step 2: Identifying Angles for Each Quadrants

In the first quadrant, the basic angle corresponding to the cosine value \(\sqrt{2}/2\) is \(\pi/4\). But, in the fourth quadrant, the corresponding angle for cosine to be \(\sqrt{2}/2\) is \(2\pi - \pi/4 = 7\pi/4\). As we are considering the angle \(x/2\), the equation becomes \((x/2) = \pi/4\) or \((x/2) = 7\pi/4\)

Step 3: Solving for \(x\)

Now, solve each equation for \(x\) using the standard value of \(\pi\) as \(3.142\). For \((x/2) = \pi/4\), \(x = 2(\pi/4) = \pi/2\). For \((x/2) = 7\pi/4\), \(x = 2(7\pi/4) = 7\pi/2\)

Key Concepts

Trigonometric Equations

Trigonometric equations are equations that involve trigonometric functions, such as sine, cosine, or tangent. Solving these equations often involves finding all angles that satisfy the given equation.
One common method to solve these equations is to employ inverse trigonometric functions or to use known trigonometric identities and special angles.
For example, an equation might involve finding the values of the angle where the trigonometric function equals a specific number, like in solving for the cosine of some angle equated to \(\sqrt{2}/2\).

• Identify the possible angles based on the trigonometric function involved.
• Take into account the periodic nature of the trigonometric functions, which can result in multiple solutions.
• Verify if the solution should be presented in degrees or radians based on context.

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It relates the angle of a right triangle to the ratio of the length of the adjacent side over the hypotenuse.
In the unit circle, the cosine of an angle is the x-coordinate of the point on the circle. This function has a periodic nature and repeats its values every \(2\pi\) radians.

• It ranges from -1 to 1, which represents all possible real values for the cosine of any angle.
• For special angles like \(\pi/4\), the cosine value is \(\sqrt{2}/2\), which is often encountered in trigonometry problems.
• Understanding how the function behaves across its domain helps in solving equations involving cosine.

Recognizing that the cosine function can have the same value in different quadrants is crucial. This concept is particularly useful when determining multiple angles that share the same cosine value.

Quadrants in Trigonometry

The trigonometric functions behave differently in each of the four quadrants of the unit circle. These quadrants help determine the sign and behavior of a trigonometric function at any given angle.

• In the first quadrant, all trigonometric functions are positive.
• In the second quadrant, sine is positive while cosine and tangent are negative.
• In the third quadrant, tangent is positive, while sine and cosine are negative.
• In the fourth quadrant, cosine is positive while sine and tangent are negative.

This knowledge is essential when solving trigonometric equations because it guides us in knowing where certain values or behaviors, like \(\cos(x/2) = \sqrt{2}/2\), might occur. In the example problem, knowing \(\cos(x/2) = \sqrt{2}/2\) can happen in the first and fourth quadrants aids in determining the correct angles.
Understanding quadrants ensures you accurately comprehend the solutions and reasons for different angle values within those quadrants, based on their signs.