Solve For { \theta$} : : : { \frac{\sin \theta}{6} - \frac{1}{8} = 0 \}

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Introduction

Trigonometric equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific type of trigonometric equation, namely the equation involving the sine function. We will use the given equation sin⁡θ6−18=0\frac{\sin \theta}{6} - \frac{1}{8} = 0 as a case study to demonstrate the step-by-step process of solving trigonometric equations.

Understanding the Equation

The given equation is sin⁡θ6−18=0\frac{\sin \theta}{6} - \frac{1}{8} = 0. To solve this equation, we need to isolate the sine function, which is the variable we are interested in. The equation involves two fractions, and we need to eliminate the fractions to simplify the equation.

Step 1: Eliminate the Fractions

To eliminate the fractions, we can multiply both sides of the equation by the least common multiple (LCM) of the denominators, which is 24. This will eliminate the fractions and simplify the equation.

\frac{\sin \theta}{6} - \frac{1}{8} = 0
\\]
24 \left( \frac{\sin \theta}{6} - \frac{1}{8} \right) = 24(0)
\\]
4 \sin \theta - 3 = 0

Step 2: Isolate the Sine Function

Now that we have eliminated the fractions, we can isolate the sine function by adding 3 to both sides of the equation. This will give us an equation with only the sine function on one side.

4 \sin \theta - 3 = 0
\\]
4 \sin \theta = 3
\\]
\sin \theta = \frac{3}{4}

Step 3: Find the Value of θ\theta

Now that we have isolated the sine function, we can find the value of θ\theta by taking the inverse sine of both sides of the equation. This will give us the angle θ\theta that satisfies the equation.

\sin \theta = \frac{3}{4}
\\]
\theta = \sin^{-1} \left( \frac{3}{4} \right)
\\]
\theta \approx 0.8486

Conclusion

In this article, we have demonstrated the step-by-step process of solving a trigonometric equation involving the sine function. We started with the given equation sin⁡θ6−18=0\frac{\sin \theta}{6} - \frac{1}{8} = 0 and eliminated the fractions by multiplying both sides by the LCM of the denominators. We then isolated the sine function by adding 3 to both sides of the equation and found the value of θ\theta by taking the inverse sine of both sides of the equation. The final answer is θ≈0.8486\theta \approx 0.8486.

Tips and Tricks

  • When solving trigonometric equations, it is essential to eliminate the fractions by multiplying both sides of the equation by the LCM of the denominators.
  • Isolate the sine function by adding or subtracting the constant term from both sides of the equation.
  • Use the inverse sine function to find the value of θ\theta that satisfies the equation.

Common Mistakes

  • Failing to eliminate the fractions by multiplying both sides of the equation by the LCM of the denominators.
  • Not isolating the sine function by adding or subtracting the constant term from both sides of the equation.
  • Not using the inverse sine function to find the value of θ\theta that satisfies the equation.

Real-World Applications

Trigonometric equations have numerous real-world applications in fields such as physics, engineering, and computer science. Some examples include:

  • Physics: Trigonometric equations are used to describe the motion of objects in terms of their position, velocity, and acceleration.
  • Engineering: Trigonometric equations are used to design and analyze electrical circuits, mechanical systems, and other engineering applications.
  • Computer Science: Trigonometric equations are used in computer graphics, game development, and other areas of computer science.

Conclusion

Introduction

In our previous article, we demonstrated the step-by-step process of solving a trigonometric equation involving the sine function. However, we understand that sometimes, it's not enough to just follow a set of instructions. You may have questions, doubts, or uncertainties about the process. That's why we've created this Q&A guide to help you better understand and apply the concepts of solving trigonometric equations.

Q: What is the first step in solving a trigonometric equation?

A: The first step in solving a trigonometric equation is to eliminate the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.

Q: How do I find the LCM of the denominators?

A: To find the LCM of the denominators, you can list the multiples of each denominator and find the smallest multiple that is common to both. Alternatively, you can use the formula: LCM(a, b) = (a × b) / GCD(a, b), where GCD is the greatest common divisor.

Q: What if the equation has multiple trigonometric functions?

A: If the equation has multiple trigonometric functions, you can use the following steps:

  1. Isolate one trigonometric function by adding or subtracting the constant term from both sides of the equation.
  2. Use the inverse function to find the value of the isolated trigonometric function.
  3. Substitute the value of the isolated trigonometric function back into the original equation.
  4. Repeat the process for each trigonometric function in the equation.

Q: How do I know which trigonometric function to use?

A: The choice of trigonometric function depends on the problem and the information given. For example, if the problem involves a right triangle, you may want to use the sine, cosine, or tangent function. If the problem involves a circular motion, you may want to use the sine or cosine function.

Q: What if I get stuck or make a mistake?

A: Don't worry! Making mistakes is a natural part of the learning process. If you get stuck or make a mistake, try the following:

  1. Review the steps you've taken so far.
  2. Check your work for errors.
  3. Ask for help from a teacher, tutor, or classmate.
  4. Take a break and come back to the problem later with a fresh perspective.

Q: How can I practice solving trigonometric equations?

A: There are many ways to practice solving trigonometric equations, including:

  1. Working through practice problems in your textbook or online resources.
  2. Using online tools or software to generate random problems.
  3. Joining a study group or finding a study buddy.
  4. Creating your own problems and solving them.

Q: What are some common mistakes to avoid when solving trigonometric equations?

A: Some common mistakes to avoid when solving trigonometric equations include:

  1. Failing to eliminate the fractions by multiplying both sides of the equation by the LCM of the denominators.
  2. Not isolating the trigonometric function by adding or subtracting the constant term from both sides of the equation.
  3. Not using the inverse function to find the value of the isolated trigonometric function.
  4. Not checking your work for errors.

Conclusion

Solving trigonometric equations can be a challenging but rewarding experience. By following the steps outlined in this Q&A guide, you can better understand and apply the concepts of solving trigonometric equations. Remember to practice regularly, ask for help when needed, and avoid common mistakes. With patience and persistence, you can become proficient in solving trigonometric equations and apply them to real-world problems.

Additional Resources

For further practice and review, we recommend the following resources:

  • Textbooks: "Trigonometry" by Michael Corral, "Trigonometry: A Unit Circle Approach" by Charles P. McKeague, and "Trigonometry" by I.M. Gelfand.
  • Online Resources: Khan Academy, MIT OpenCourseWare, and Wolfram Alpha.
  • Software: GeoGebra, Mathway, and Wolfram Alpha.

Final Tips

  • Practice regularly to build your skills and confidence.
  • Ask for help when needed, whether it's from a teacher, tutor, or classmate.
  • Review and practice regularly to reinforce your understanding.
  • Apply trigonometric equations to real-world problems to see the relevance and importance of the subject.