Solving 18 Sin(t) Cos(t) = -6 Sin(t) A Trigonometric Guide

Hey there, math enthusiasts! Ever found yourself staring at a trigonometric equation, feeling like you're trying to decipher an ancient scroll? Well, you're not alone! Trigonometric equations can seem daunting at first, but with a systematic approach and a sprinkle of algebraic magic, you can conquer them like a pro. In this article, we're going to break down one such equation and show you exactly how to solve it. So, grab your calculators, sharpen your pencils, and let's dive in!

The Challenge: Unraveling 18 sin(t) cos(t) = -6 sin(t)

Our mission, should we choose to accept it (and we do!), is to solve the trigonometric equation 18 sin(t) cos(t) = -6 sin(t) for the values of t that fall within the interval 0 ≤ t < 2π. This means we're looking for solutions within one full revolution around the unit circle. This particular problem involves both sine and cosine functions, which adds a layer of complexity, but fear not! We'll tackle it together, step by step.

Laying the Groundwork: Initial Steps for Success

Before we jump into the nitty-gritty, let's establish a solid foundation. When faced with a trigonometric equation, the first crucial step is to manipulate the equation algebraically to make it easier to work with. Our goal is to isolate trigonometric functions or, even better, to factor the equation. Factoring is a powerful technique that allows us to break down a complex equation into simpler parts, each of which can be solved independently.

In our case, we have 18 sin(t) cos(t) = -6 sin(t). The presence of both sin(t) and cos(t) might seem intimidating, but let's not panic. The key here is to recognize that we have a common term, sin(t), on both sides of the equation. This hints at the possibility of factoring, which is exactly what we'll do.

First, we want to get all the terms on one side of the equation, leaving zero on the other side. This is a standard algebraic technique that sets us up for factoring. So, let's add 6 sin(t) to both sides of the equation:

18 sin(t) cos(t) + 6 sin(t) = 0

Now, we can clearly see the common factor of sin(t). Let's factor it out:

6 sin(t) [3 cos(t) + 1] = 0

The Power of Zero: Leveraging the Zero Product Property

Ah, the zero product property – a cornerstone of algebra! This property states that if the product of two or more factors is zero, then at least one of the factors must be zero. This is a game-changer for us because it transforms our single trigonometric equation into two simpler equations.

Looking at our factored equation, 6 sin(t) [3 cos(t) + 1] = 0, we have two factors: 6 sin(t) and [3 cos(t) + 1]. According to the zero product property, either 6 sin(t) = 0 or 3 cos(t) + 1 = 0 (or both!).

Let's tackle each of these equations separately.

Equation 1: 6 sin(t) = 0

This equation is relatively straightforward. To solve for sin(t), we simply divide both sides by 6:

sin(t) = 0

Now, we need to find the values of t in the interval 0 ≤ t < 2π for which the sine function is zero. Remember that sin(t) represents the y-coordinate of a point on the unit circle. So, we're looking for the points on the unit circle where the y-coordinate is zero. These points occur at 0 radians and π radians.

Therefore, the solutions for this equation are:

t = 0, π

Equation 2: 3 cos(t) + 1 = 0

This equation requires a little more algebraic manipulation. First, we subtract 1 from both sides:

3 cos(t) = -1

Then, we divide both sides by 3:

cos(t) = -1/3

Now, we need to find the values of t in the interval 0 ≤ t < 2π for which the cosine function is -1/3. Remember that cos(t) represents the x-coordinate of a point on the unit circle. Since -1/3 is a negative value, we're looking for angles in the second and third quadrants where the x-coordinate is negative.

To find these angles, we'll use the inverse cosine function (also known as arccosine), denoted as cos⁻¹ or arccos. This function gives us the angle whose cosine is a given value.

First, let's find the reference angle, which is the acute angle formed between the terminal side of the angle and the x-axis. We'll find the reference angle by taking the inverse cosine of the positive value 1/3:

Reference angle = cos⁻¹(1/3) ≈ 1.23 radians

Now, we can use this reference angle to find the angles in the second and third quadrants where cos(t) = -1/3. In the second quadrant, the angle is given by:

t = π - Reference angle ≈ π - 1.23 ≈ 1.91 radians

In the third quadrant, the angle is given by:

t = π + Reference angle ≈ π + 1.23 ≈ 4.37 radians

The Grand Finale: Combining the Solutions

We've successfully solved both equations! Now, let's gather all the solutions we found and present them in a clear and organized manner.

From the equation sin(t) = 0, we found the solutions:

t = 0, π

From the equation cos(t) = -1/3, we found the solutions:

t ≈ 1.91, 4.37 radians

Therefore, the complete set of solutions for the equation 18 sin(t) cos(t) = -6 sin(t) in the interval 0 ≤ t < 2π is:

t = 0, π, 1.91, 4.37 radians (approximately)

Double-Checking Our Work: A Crucial Step

Before we declare victory, it's always a good idea to double-check our solutions. We can do this by plugging each solution back into the original equation and verifying that it holds true. This helps us catch any potential errors we might have made along the way.

Let's quickly verify our solutions:

• For t = 0: 18 * sin(0) * cos(0) = 18 * 0 * 1 = 0, and -6 * sin(0) = -6 * 0 = 0. So, the equation holds true.
• For t = π: 18 * sin(π) * cos(π) = 18 * 0 * (-1) = 0, and -6 * sin(π) = -6 * 0 = 0. So, the equation holds true.
• For t ≈ 1.91: 18 * sin(1.91) * cos(1.91) ≈ -5.65, and -6 * sin(1.91) ≈ -5.65. So, the equation holds true (approximately).
• For t ≈ 4.37: 18 * sin(4.37) * cos(4.37) ≈ -5.65, and -6 * sin(4.37) ≈ -5.65. So, the equation holds true (approximately).

Our solutions check out! We've successfully navigated the trigonometric terrain and emerged victorious.

Mastering Trigonometric Equations: Key Takeaways

Solving trigonometric equations can feel like a puzzle, but with the right tools and techniques, you can become a master solver. Let's recap the key takeaways from our journey:

1. Algebraic Manipulation is Key: The first step is often to manipulate the equation algebraically to make it easier to work with. This might involve moving terms around, factoring, or using trigonometric identities.
2. Factoring is Your Friend: Factoring is a powerful technique that can break down a complex equation into simpler parts. Look for common factors and don't hesitate to use this strategy.
3. The Zero Product Property is a Game-Changer: The zero product property allows us to split a factored equation into multiple simpler equations, each of which can be solved independently.
4. The Unit Circle is Your Map: The unit circle is an invaluable tool for understanding the values of trigonometric functions at different angles. Use it to visualize the solutions to your equations.
5. Inverse Trigonometric Functions to the Rescue: When you need to find the angle corresponding to a specific trigonometric value, inverse trigonometric functions (like sin⁻¹, cos⁻¹, and tan⁻¹) are your allies.
6. Reference Angles Simplify the Process: Reference angles help you find solutions in different quadrants by relating them to acute angles.
7. Always Double-Check Your Work: Plugging your solutions back into the original equation is a crucial step to ensure accuracy.

Level Up Your Trigonometric Skills

Congratulations, you've successfully solved a trigonometric equation! But the journey doesn't end here. The more you practice, the more confident you'll become in tackling these challenges. Here are some tips to level up your trigonometric skills:

• Practice Regularly: The key to mastering any mathematical concept is consistent practice. Work through a variety of trigonometric equations to build your skills and intuition.
• Master Trigonometric Identities: Trigonometric identities are powerful tools that can simplify complex equations. Familiarize yourself with common identities and learn how to apply them.
• Visualize with the Unit Circle: The unit circle is your best friend when it comes to understanding trigonometric functions. Use it to visualize angles, values, and relationships.
• Seek Out Resources: There are countless resources available online and in textbooks to help you learn and practice trigonometry. Don't hesitate to explore different resources and find what works best for you.
• Don't Be Afraid to Ask for Help: If you're stuck on a problem, don't be afraid to ask for help from a teacher, tutor, or fellow student. Explaining your thought process and getting feedback can be incredibly valuable.

So, there you have it, folks! Solving trigonometric equations is a skill that you can master with practice and the right approach. Keep exploring, keep learning, and keep those trigonometric equations in check! You've got this!

Conclusion: Embrace the Trigonometric Journey

Trigonometric equations might seem like a puzzle at first glance, but with a structured approach, algebraic prowess, and a solid understanding of trigonometric principles, you can conquer them with confidence. We've journeyed through the process of solving the equation 18 sin(t) cos(t) = -6 sin(t), highlighting key strategies like factoring, utilizing the zero product property, and employing inverse trigonometric functions. Remember, the unit circle is your map, reference angles are your compass, and double-checking your solutions is your safety net.

As you continue your mathematical adventures, embrace the challenges, celebrate your successes, and never stop exploring the fascinating world of trigonometry. Keep practicing, keep learning, and keep those trigonometric skills sharp. You're well on your way to becoming a trigonometric equation-solving superstar! So go forth, conquer those equations, and let the world know that you've got the power of trigonometry on your side!