Solve For \[$\theta\$\]:$\[4 \cos^2 \theta + 4 \sin \theta = 5 \quad \left(0^{\circ} \leq \theta \leq 90^{\circ}\right)\\]

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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 involving the cosine and sine functions. We will use the given equation $4 \cos^2 \theta + 4 \sin \theta = 5$ to demonstrate the step-by-step process of solving trigonometric equations.

Understanding the Equation

The given equation is $4 \cos^2 \theta + 4 \sin \theta = 5$. This equation involves the cosine and sine functions, which are periodic functions that oscillate between -1 and 1. The equation is also restricted to the interval $0^{\circ} \leq \theta \leq 90^{\circ}$, which means we are only interested in finding the value of $\theta$ within this range.

Step 1: Simplify the Equation

To simplify the equation, we can start by factoring out the common term $4$ from the left-hand side of the equation.

$$4 \cos^2 \theta + 4 \sin \theta = 5$$

$$4(\cos^2 \theta + \sin \theta) = 5$$

Now, we can divide both sides of the equation by $4$ to isolate the term inside the parentheses.

$$\cos^2 \theta + \sin \theta = \frac{5}{4}$$

Step 2: Use Trigonometric Identities

We can use the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ to rewrite the equation.

$$\cos^2 \theta + \sin \theta = \frac{5}{4}$$

$$\sin^2 \theta + \cos^2 \theta + \sin \theta = \frac{5}{4} + \sin \theta$$

Using the Pythagorean identity, we can substitute $\sin^2 \theta + \cos^2 \theta$ with $1$.

$$1 + \sin \theta = \frac{5}{4} + \sin \theta$$

Now, we can subtract $\sin \theta$ from both sides of the equation to isolate the term.

$$1 = \frac{5}{4}$$

This is a contradiction, which means that the original equation has no solution.

Conclusion

In this article, we demonstrated the step-by-step process of solving a trigonometric equation involving the cosine and sine functions. We started by simplifying the equation, using trigonometric identities, and finally, we found that the original equation has no solution. This example highlights the importance of carefully analyzing the equation and using the appropriate trigonometric identities to solve it.

Tips and Tricks

• When solving trigonometric equations, it's essential to carefully analyze the equation and identify the type of trigonometric function involved.
• Use trigonometric identities to simplify the equation and make it easier to solve.
• Pay attention to the restrictions on the variable, such as the interval $0^{\circ} \leq \theta \leq 90^{\circ}$ in this example.

Common Mistakes to Avoid

• Failing to simplify the equation using trigonometric identities.
• Not paying attention to the restrictions on the variable.
• Not carefully analyzing the equation to identify the type of trigonometric function involved.

Real-World Applications

Trigonometric equations have numerous real-world applications in fields such as physics, engineering, and computer science. For example, trigonometric equations are used to model the motion of objects, such as the trajectory of a projectile or the vibration of a spring.

Conclusion

In conclusion, solving trigonometric equations is a crucial skill for students and professionals alike. By following the step-by-step process outlined in this article, you can develop the skills and confidence to tackle even the most challenging trigonometric equations. Remember to carefully analyze the equation, use trigonometric identities, and pay attention to the restrictions on the variable. With practice and patience, you can become proficient in solving trigonometric equations and apply your skills to real-world problems.

Additional Resources

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

• Khan Academy: Trigonometry
• MIT OpenCourseWare: Trigonometry
• Wolfram Alpha: Trigonometric Equations

Final Thoughts

Q: What is a trigonometric equation?

A: A trigonometric equation is an equation that involves trigonometric functions, such as sine, cosine, and tangent. These equations can be used to model real-world problems, such as the motion of objects or the vibration of springs.

Q: What are the common trigonometric functions?

A: The common trigonometric functions are:

• Sine (sin)
• Cosine (cos)
• Tangent (tan)
• Cotangent (cot)
• Secant (sec)
• Cosecant (csc)

Q: How do I solve a trigonometric equation?

A: To solve a trigonometric equation, follow these steps:

1. Simplify the equation using trigonometric identities.
2. Isolate the trigonometric function.
3. Use algebraic techniques to solve for the variable.
4. Check the solution to ensure it satisfies the original equation.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

• Pythagorean identity: sin^2(x) + cos^2(x) = 1
• Sum and difference identities: sin(x + y) = sin(x)cos(y) + cos(x)sin(y)
• Double-angle identities: sin(2x) = 2sin(x)cos(x)
• Half-angle identities: sin(x/2) = sqrt((1 - cos(x))/2)

Q: How do I use trigonometric identities to simplify an equation?

A: To use trigonometric identities to simplify an equation, follow these steps:

1. Identify the trigonometric function involved in the equation.
2. Choose the appropriate trigonometric identity to apply.
3. Substitute the identity into the equation.
4. Simplify the equation using algebraic techniques.

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

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

• Failing to simplify the equation using trigonometric identities.
• Not isolating the trigonometric function.
• Not checking the solution to ensure it satisfies the original equation.
• Not paying attention to the restrictions on the variable.

Q: How do I check my solution to ensure it satisfies the original equation?

A: To check your solution, substitute the value of the variable back into the original equation and simplify. If the equation is true, then the solution is correct.

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

A: Trigonometric equations have numerous real-world applications, including:

• Modeling the motion of objects, such as the trajectory of a projectile or the vibration of a spring.
• Analyzing the behavior of electrical circuits.
• Solving problems in physics, engineering, and computer science.

Q: How can I practice solving trigonometric equations?

A: To practice solving trigonometric equations, try the following:

• Use online resources, such as Khan Academy or MIT OpenCourseWare, to practice solving trigonometric equations.
• Work through practice problems in a textbook or online resource.
• Join a study group or find a study partner to practice solving trigonometric equations together.

Q: What are some additional resources for learning trigonometric equations?

A: Some additional resources for learning trigonometric equations include:

• Khan Academy: Trigonometry
• MIT OpenCourseWare: Trigonometry
• Wolfram Alpha: Trigonometric Equations
• Online textbooks and resources, such as Mathway or Symbolab.