Problem 21 - Algebra and Trigonometry

Question

\(17-24\) n Solve the given equation, and list six specific solutions. $$ \cos \theta=0.28 $$

Step-by-Step Solution

Answer

Specific solutions include \( \theta = 73.74^\circ, 433.74^\circ, -286.26^\circ, 793.74^\circ, -646.26^\circ, 1153.74^\circ \).

Step 1: Understand the Equation

The given equation is \( \cos \theta = 0.28 \). We need to find the angles \( \theta \) within a specified range where the cosine of \( \theta \) equals 0.28.

Step 2: Use the Inverse Cosine Function

To find the primary solution for \( \theta \), use the inverse cosine function: \( \theta = \cos^{-1}(0.28) \). Calculate this using a calculator or appropriate software.

Step 3: Calculate the Primary Solution

Using a calculator, compute \( \cos^{-1}(0.28) \) to find one specific solution. This value is approximately \( \theta = 73.74^\circ \) (rounded to two decimal places).

Step 4: Identify Periodicity of Cosine Function

The cosine function has a period of \( 360^\circ \). Therefore, we can find additional solutions by adding integer multiples of \( 360^\circ \) to the primary solution \( \theta = 73.74^\circ \), resulting in solutions such as: \( \theta = 73.74^\circ + n\cdot 360^\circ \), where \( n \) is an integer.

Step 5: Find Specific Solutions

We need to list six specific solutions. Starting with \( \theta = 73.74^\circ \), calculate additional solutions by adding and subtracting \( 360^\circ \):1. \( \theta = 73.74^\circ \)2. \( \theta = 73.74^\circ + 360^\circ = 433.74^\circ \)3. \( \theta = 73.74^\circ - 360^\circ = -286.26^\circ \)4. \( \theta = 73.74^\circ + 720^\circ = 793.74^\circ \)5. \( \theta = 73.74^\circ - 720^\circ = -646.26^\circ \)6. \( \theta = 73.74^\circ + 1080^\circ = 1153.74^\circ \)

Step 6: Verify Solutions

Check that each calculated \( \theta \) satisfies the condition \( \cos \theta = 0.28 \). Use a calculator to confirm the cosine of each angle is approximately 0.28.

Key Concepts

Inverse Cosine Function

The inverse cosine function, often denoted as \( \cos^{-1} \) or \( \text{acos} \), is used to determine the angle whose cosine is a specific value. When you solve an equation like \( \cos \theta = 0.28 \), you're essentially asking: "What angle \( \theta \) has a cosine of 0.28?"

To find this angle, you use the inverse cosine function: \( \theta = \cos^{-1}(0.28) \). This function is typically restricted to the range of 0 to 180 degrees (or \([0, \pi]\) in radians) because the cosine function is one-to-one within this interval. This means for every x value, there's exactly one y value.

• Calculator or Software: Use a calculator or software capable of trigonometric functions to compute \( \cos^{-1}(0.28) \).
• Result: The primary solution is approximately \( 73.74^\circ \).

Understanding how to use the inverse cosine function is crucial for finding angles that solve trigonometric equations.

Periodicity of Trigonometric Functions

Trigonometric functions, including the cosine function, display a property known as periodicity. This means they repeat their values in regular intervals. For cosine, this interval is \( 360^\circ \) or \( 2\pi \) radians. Put simply, adding or subtracting full cycles (multiples of \( 360^\circ \)) from an angle does not change its cosine value.

In the context of solving the equation \( \cos \theta = 0.28 \), once you find an initial solution, like \( \theta = 73.74^\circ \), you can find additional solutions by adding or subtracting \( 360^\circ \). For example:

• \( \theta = 73.74^\circ + 360^\circ = 433.74^\circ \)
• \( \theta = 73.74^\circ - 360^\circ = -286.26^\circ \)

This principle allows you to easily generate multiple solutions for a trigonometric equation by recognizing the periodicity involved.

Specific Solutions of Trigonometric Equations

Finding specific solutions in trigonometric equations, like \( \cos \theta = 0.28 \), involves identifying angles that satisfy the equation. Once you've determined the primary solution using \( \cos^{-1} \), you can utilize the periodicity of cosine to generate more solutions.

To achieve this:

• Start with the initial solution: \( \theta = 73.74^\circ \)
• Add and subtract multiples of \( 360^\circ \) to find more angles:
• \( \theta = 73.74^\circ + 360^\circ = 433.74^\circ \)
• \( \theta = 73.74^\circ - 360^\circ = -286.26^\circ \)
• Continue, adding \( 720^\circ \) for further solutions: \( \theta = 793.74^\circ \), etc.

Use a calculator to verify that each angle's cosine is indeed 0.28. This ensures the accuracy of your solutions and their fit within the problem's requirements.