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Chapter 7: Problem 10

\(\cos ^{3} \theta+.47 \cos \theta-1.23=0, \quad 0<\theta<90^{\circ}\)

Short Answer

Expert verified
Question: Solve the equation \(\cos^3 \theta + 0.47\cos \theta - 1.23 = 0\) for \(\theta\) within the interval \(0<\theta<90^{\circ}\). Answer: To solve the equation, first rewrite it in terms of the variable \(x\) such that \(x = \cos \theta\), giving us the equation \(x^3 + 0.47x - 1.23 = 0\). Then, use a numerical method such as an online calculator to find the real root \(x_0\). Find the corresponding angle \(\theta\) by taking the inverse cosine (arccos) of \(x_0\), i.e., \(\theta = \arccos(x_0)\). Finally, ensure that the angle falls within the specified interval \(0<\theta<90^{\circ}\). If it does, then you have found a valid solution for \(\theta\).

Step by step solution

01

1. Rewrite the equation for better understanding.

We are given the equation \(\cos^3 \theta + 0.47\cos \theta - 1.23 = 0\). Let's denote \(x = \cos \theta\). So, the equation can be rewritten as \(x^3 + 0.47x - 1.23 = 0\).
02

2. Solve the equation for x using a numerical method.

The equation \(x^3 + 0.47x - 1.23 = 0\) cannot be easily factored, so we can use a numerical method such as the bisection method or the Newton-Raphson method to find the approximate value(s) of \(x\) that satisfy the equation. For simplicity, we will use an online calculator or software to solve this equation numerically. Doing so, we find that there is one real root for \(x\) within the range \(-1 \leq x \leq 1\). Let's call this root \(x_0\).
03

3. Find the corresponding angle for the x value.

Having found the value of \(x_0\), we can now find the corresponding angle \(\theta\). Recall that we denoted \(x = \cos \theta\) at the beginning of the problem. So, we have \(x_0 = \cos \theta\). Now, in order to find the angle \(\theta\), we can take the inverse cosine (also known as arccos) of both sides of the equation: \(\theta = \arccos(x_0)\).
04

4. Check if the angle falls within the specified range.

We have found the angle \(\theta = \arccos(x_0)\), but we still need to make sure it falls within the specified interval \(0<\theta<90^{\circ}\). You can find this out by calculating the value of the arccos function and ensuring that it lies within the interval \(0<\theta<90^{\circ}\) or, equivalently, \(0<\theta< \frac{\pi}{2}\) radians. If the angle \(\theta\) is indeed within this interval, then we have found a valid solution to the given equation.
05

5. Conclusion

In conclusion, we have solved the given equation \(\cos^3 \theta + 0.47 \cos \theta -1.23 = 0\) for the angle \(\theta\) within the interval \(0<\theta<90^{\circ}\) by rewriting the equation in terms of the variable \(x\), solving numerically for the real root \(x_0\), finding the corresponding angle \(\theta\) using the inverse cosine function, and checking if the angle falls within the specified interval.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Numerical Methods
Numerical methods are powerful tools used to approximate solutions for equations that cannot be easily solved algebraically. In this exercise, since the equation \(x^3 + 0.47x - 1.23 = 0\) does not lend itself to straightforward factoring, we employ a numerical method to find the roots. Common methods include:
  • Bisection Method: This method involves repeatedly dividing an interval in half and selecting the subinterval in which the root lies, based on the signs of the function at the endpoints.
  • Newton-Raphson Method: This method uses the derivative of the function to find successively better approximations to the roots of a real-valued function.
Given that the context here involves a cosine equation, we use software tools or calculators to streamline this process and calculate the root of the function. This automated numerical method resulted in a real root within the interval \(-1 \leq x \leq 1\).
Inverse Trigonometric Functions
Inverse trigonometric functions are used to find angles when the value of the trigonometric function is known. In our problem, once the value of \(x\) is found where \(x = \cos \theta\), we need to determine the angle \(\theta\). This is where the inverse cosine function, or \(\arccos\), is employed.

The function \(\arccos(x)\) returns an angle whose cosine is \(x\). Therefore, once we have determined \(x_0\) through numerical methods, we find \(\theta = \arccos(x_0)\). This concept is crucial for problems involving trigonometric equations, as it bridges the value found numerically to a meaningful angle measurement.
Problem-Solving Steps
Problem-solving steps provide a structured approach to tackle equations systematically. The given problem requires us to proceed through specific stages, ensuring that each part is thoroughly addressed:
  • Step 1: Rewrite the equation using a substitution to simplify it.
  • Step 2: Use numerical methods to solve the simplified equation for \(x\).
  • Step 3: Upon finding \(x\), use the inverse cosine function to find \(\theta\).
  • Step 4: Verify that the resultant angle falls within the required domain \(0 < \theta < 90^\circ\).
Each of these steps ensures clarity and correctness in solving trigonometric equations effectively.
Angle Measurement in Degrees
Angle measurement is an essential component of solving trigonometric equations within a specified interval. In this problem, we are concerned with angles measured in degrees, ranging from \(0\) to \(90^\circ\).

This range corresponds to the first quadrant of the unit circle where the cosine function is non-negative. Ensuring that \(\theta\) falls within this range confirms the solution's validity. It's important to remember the conversion between degrees and radians, noting that \(90^\circ\) is equivalent to \(\frac{\pi}{2}\) radians, which helps in understanding the angle's position on the unit circle.

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