Solve The Equation: $\[ 3 - \sin \theta = \cos (2 \theta) \\]

Introduction

Trigonometric equations are a fundamental concept in mathematics, and solving them requires a deep understanding of trigonometric functions and their properties. In this article, we will focus on solving the equation $3 - \sin \theta = \cos (2 \theta)$, which involves the use of trigonometric identities and techniques.

Understanding the Equation

The given equation is $3 - \sin \theta = \cos (2 \theta)$. To solve this equation, we need to simplify it and express it in terms of a single trigonometric function. We can start by using the double-angle identity for cosine, which states that $\cos (2 \theta) = 1 - 2 \sin^2 \theta$.

Step 1: Simplify the Equation

Using the double-angle identity for cosine, we can rewrite the equation as:

$$3 - \sin \theta = 1 - 2 \sin^2 \theta$$

Now, we can simplify the equation by combining like terms:

$$2 \sin^2 \theta + \sin \theta - 2 = 0$$

This is a quadratic equation in terms of $\sin \theta$, and we can solve it using the quadratic formula.

Step 2: Solve the Quadratic Equation

The quadratic formula states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In our case, $a = 2$, $b = 1$, and $c = -2$. Plugging these values into the quadratic formula, we get:

$$\sin \theta = \frac{-1 \pm \sqrt{1 + 16}}{4}$$

Simplifying further, we get:

$$\sin \theta = \frac{-1 \pm \sqrt{17}}{4}$$

Step 3: Find the Values of $\theta$

Now that we have the values of $\sin \theta$, we can find the corresponding values of $\theta$ using the inverse sine function.

$$\theta = \sin^{-1} \left( \frac{-1 \pm \sqrt{17}}{4} \right)$$

Using a calculator or a trigonometric table, we can find the values of $\theta$ that satisfy the equation.

Step 4: Check the Solutions

Once we have found the values of $\theta$, we need to check if they satisfy the original equation. We can do this by plugging the values of $\theta$ back into the original equation and checking if it holds true.

Conclusion

Solving the equation $3 - \sin \theta = \cos (2 \theta)$ requires a deep understanding of trigonometric functions and their properties. By using the double-angle identity for cosine and the quadratic formula, we can simplify the equation and find the values of $\theta$ that satisfy it. However, it's essential to check the solutions to ensure that they satisfy the original equation.

Tips and Tricks

• When solving trigonometric equations, it's essential to use the correct trigonometric identities and techniques.
• Always check the solutions to ensure that they satisfy the original equation.
• Use a calculator or a trigonometric table to find the values of $\theta$ that satisfy the equation.

Common Mistakes

• Failing to use the correct trigonometric identities and techniques.
• Not checking the solutions to ensure that they satisfy the original equation.
• Using the wrong values of $\theta$ or not using the correct units.

Real-World Applications

Trigonometric equations have numerous real-world applications in fields such as physics, engineering, and computer science. For example, they can be used to model the motion of objects, describe the behavior of electrical circuits, and optimize the performance of algorithms.

Further Reading

For further reading on trigonometric equations, we recommend the following resources:

• "Trigonometry" by Michael Corral
• "Calculus" by Michael Spivak
• "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Conclusion

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Introduction

In our previous article, we discussed how to solve the equation $3 - \sin \theta = \cos (2 \theta)$ using trigonometric identities and techniques. However, we know that solving trigonometric equations can be a challenging task, and many students struggle with it. In this article, we will provide a Q&A guide to help you better understand how to solve trigonometric equations.

Q: What are the most common trigonometric identities used to solve equations?

A: The most common trigonometric identities used to solve equations are:

• Double-angle identity for sine: $\sin (2 \theta) = 2 \sin \theta \cos \theta$
• Double-angle identity for cosine: $\cos (2 \theta) = 1 - 2 \sin^2 \theta$
• Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$

Q: How do I simplify a trigonometric equation?

A: To simplify a trigonometric equation, you can use the following steps:

1. Use the double-angle identity for sine or cosine: If the equation contains a double-angle trigonometric function, try to simplify it using the double-angle identity.
2. Use the Pythagorean identity: If the equation contains a trigonometric function and its square, try to simplify it using the Pythagorean identity.
3. Combine like terms: Combine the like terms in the equation to simplify it further.

Q: How do I solve a quadratic equation in terms of a trigonometric function?

A: To solve a quadratic equation in terms of a trigonometric function, you can use the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this formula, $a$, $b$, and $c$ are the coefficients of the quadratic equation, and $x$ is the variable.

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

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

• Failing to use the correct trigonometric identities and techniques: Make sure to use the correct trigonometric identities and techniques to simplify the equation.
• Not checking the solutions: Always check the solutions to ensure that they satisfy the original equation.
• Using the wrong values of $\theta$ or not using the correct units: Make sure to use the correct values of $\theta$ and units when solving the equation.

Q: How do I check the solutions to a trigonometric equation?

A: To check the solutions to a trigonometric equation, you can plug the values of $\theta$ back into the original equation and check if it holds true.

Q: What are some real-world applications of trigonometric equations?

A: Trigonometric equations have numerous real-world applications in fields such as physics, engineering, and computer science. For example, they can be used to model the motion of objects, describe the behavior of electrical circuits, and optimize the performance of algorithms.

Q: Where can I find more resources on trigonometric equations?

A: You can find more resources on trigonometric equations in the following places:

• Textbooks: "Trigonometry" by Michael Corral, "Calculus" by Michael Spivak, and "Mathematics for Computer Science" by Eric Lehman and Tom Leighton.
• Online resources: Khan Academy, MIT OpenCourseWare, and Wolfram Alpha.
• Practice problems: Try solving practice problems on websites such as Mathway and Symbolab.

Conclusion

Solving trigonometric equations can be a challenging task, but with the right techniques and resources, you can master it. Remember to use the correct trigonometric identities and techniques, check the solutions, and use the correct values of $\theta$ and units. With practice and patience, you can become proficient in solving trigonometric equations and apply them to real-world problems.