Solve $12 \cos^2(t) + 7 \sin(t) - 13 = 0$ For All Solutions Where $0 \leq T \ \textless \ 2\pi$. $t =$

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 trigonometric equation, $12 \cos^2(t) + 7 \sin(t) - 13 = 0$, for all solutions where $0 \leq t \ \textless \ 2\pi$. We will break down the solution process into manageable steps, making it easier to understand and apply.

Understanding the Equation

The given equation is a quadratic equation in terms of $\cos(t)$ and $\sin(t)$. To solve it, we need to manipulate the equation to isolate the trigonometric functions. The equation can be rewritten as:

$$12 \cos^2(t) + 7 \sin(t) - 13 = 0$$

Our goal is to find the values of $t$ that satisfy this equation.

Step 1: Isolate the Trigonometric Functions

To isolate the trigonometric functions, we can start by rearranging the equation:

$$12 \cos^2(t) = -7 \sin(t) + 13$$

Next, we can divide both sides by 12:

$$\cos^2(t) = \frac{-7 \sin(t) + 13}{12}$$

Step 2: Use the Pythagorean Identity

The Pythagorean identity states that $\sin^2(t) + \cos^2(t) = 1$. We can use this identity to rewrite the equation:

$$\cos^2(t) = 1 - \sin^2(t)$$

Substituting this expression into the previous equation, we get:

$$1 - \sin^2(t) = \frac{-7 \sin(t) + 13}{12}$$

Step 3: Simplify the Equation

To simplify the equation, we can multiply both sides by 12:

$$12 - 12 \sin^2(t) = -7 \sin(t) + 13$$

Next, we can rearrange the equation to get:

$$12 \sin^2(t) + 7 \sin(t) - 5 = 0$$

Step 4: Solve the Quadratic Equation

The equation is a quadratic equation in terms of $\sin(t)$. We can solve it using the quadratic formula:

$$\sin(t) = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, $a = 12$, $b = 7$, and $c = -5$. Plugging these values into the formula, we get:

$$\sin(t) = \frac{-7 \pm \sqrt{7^2 - 4(12)(-5)}}{2(12)}$$

Simplifying the expression, we get:

$$\sin(t) = \frac{-7 \pm \sqrt{49 + 240}}{24}$$

$$\sin(t) = \frac{-7 \pm \sqrt{289}}{24}$$

$$\sin(t) = \frac{-7 \pm 17}{24}$$

This gives us two possible values for $\sin(t)$:

$$\sin(t) = \frac{-7 + 17}{24} = \frac{10}{24} = \frac{5}{12}$$

$$\sin(t) = \frac{-7 - 17}{24} = \frac{-24}{24} = -1$$

Step 5: Find the Values of $t$

Now that we have the values of $\sin(t)$, we can find the corresponding values of $t$. We will use the inverse sine function to find the values of $t$.

For $\sin(t) = \frac{5}{12}$, we have:

$$t = \sin^{-1}\left(\frac{5}{12}\right)$$

Using a calculator or a trigonometric table, we can find the value of $t$:

$$t \approx 0.394$$

For $\sin(t) = -1$, we have:

$$t = \sin^{-1}(-1)$$

Using a calculator or a trigonometric table, we can find the value of $t$:

$$t = \frac{3\pi}{2}$$

Conclusion

In this article, we solved the trigonometric equation $12 \cos^2(t) + 7 \sin(t) - 13 = 0$ for all solutions where $0 \leq t \ \textless \ 2\pi$. We broke down the solution process into manageable steps, making it easier to understand and apply. We used the Pythagorean identity, simplified the equation, and solved the quadratic equation to find the values of $t$. The final values of $t$ are:

$$t \approx 0.394$$

$$t = \frac{3\pi}{2}$$

These values satisfy the given equation and are within the specified range.

Final Answer

The final answer is:

t \approx 0.394, t = \frac{3\pi}{2}$<br/> **Solving Trigonometric Equations: A Q&A Guide** ===================================================== **Introduction** --------------- In our previous article, we solved the trigonometric equation $12 \cos^2(t) + 7 \sin(t) - 13 = 0$ for all solutions where $0 \leq t \ \textless \ 2\pi$. In this article, we will provide a Q&A guide to help you better understand the solution process and address any questions you may have. **Q: What is the first step in solving a trigonometric equation?** --------------------------------------------------------- A: The first step in solving a trigonometric equation is to isolate the trigonometric functions. This involves rearranging the equation to get all the trigonometric functions on one side. **Q: How do I use the Pythagorean identity in solving a trigonometric equation?** -------------------------------------------------------------------------------- A: The Pythagorean identity states that $\sin^2(t) + \cos^2(t) = 1$. You can use this identity to rewrite the equation and simplify it. **Q: What is the quadratic formula, and how do I use it to solve a quadratic equation?** ----------------------------------------------------------------------------------------- A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

You can use this formula to solve a quadratic equation by plugging in the values of $a$, $b$, and $c$.

Q: How do I find the values of $t$ that satisfy a trigonometric equation?

A: To find the values of $t$ that satisfy a trigonometric equation, you can use the inverse trigonometric functions. For example, if you have an equation involving $\sin(t)$, you can use the inverse sine function to find the value of $t$.

Q: What are some common trigonometric identities that I should know?

A: Some common trigonometric identities that you should know include:

• $\sin^2(t) + \cos^2(t) = 1$
• $\tan(t) = \frac{\sin(t)}{\cos(t)}$
• $\cot(t) = \frac{\cos(t)}{\sin(t)}$

Q: How do I simplify a trigonometric expression?

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

1. Combine like terms
2. Use the Pythagorean identity to rewrite the expression
3. Simplify the expression using algebraic manipulations

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

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

• Not isolating the trigonometric functions
• Not using the Pythagorean identity
• Not simplifying the expression
• Not checking the solutions

Conclusion

In this article, we provided a Q&A guide to help you better understand the solution process for solving trigonometric equations. We covered topics such as isolating trigonometric functions, using the Pythagorean identity, and simplifying expressions. We also discussed common mistakes to avoid and provided some common trigonometric identities that you should know.

Final Tips

• Practice solving trigonometric equations to become more comfortable with the solution process.
• Use the Pythagorean identity to simplify expressions and solve equations.
• Check your solutions to ensure that they satisfy the original equation.
• Use the inverse trigonometric functions to find the values of $t$ that satisfy a trigonometric equation.

By following these tips and practicing regularly, you will become more confident and proficient in solving trigonometric equations.