Solving $6 ext{sin}^2( heta) + ext{sin}( heta) - 5 = 0$: Find $\theta$

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Hey guys! Today, we're diving deep into the world of trigonometry to tackle a fascinating problem: solving the equation $6 ext{sin}^2(\theta) + ext{sin}(\theta) - 5 = 0$. We’re aiming to find all the angles $\theta$ within the range of $0^{\circ}$ to $360^{\circ}$ that make this equation true. We'll break it down step-by-step, so even if you find trig equations a bit intimidating, you'll walk away with a solid understanding. So, buckle up and let's get started!

Understanding the Problem

Before we jump into solving, let's make sure we understand what the question is asking. We have a trigonometric equation involving the sine function, $\text{sin}(\theta)$. Our mission is to find all possible values of the angle $\theta$, between $0^{\circ}$ and $360^{\circ}$, that satisfy the equation $6 \text{sin}^2(\theta) + \text{sin}(\theta) - 5 = 0$. Think of it like this: we're looking for the angles that, when plugged into the equation, make both sides equal. And we need to round our final answers to the nearest tenth of a degree. Easy peasy, right?

Why is this important?

Trigonometric equations pop up all over the place in the real world. From physics and engineering to navigation and even music, understanding how to solve these equations is super useful. Plus, mastering trig equations is a fundamental skill for anyone studying math or science at a higher level.

Step-by-Step Solution

Okay, let's get down to business and solve this equation. We're going to use a clever trick to make it look more familiar, and then we'll use our algebra skills to find the solutions.

1. Recognize the Quadratic Form

Notice that the equation $6 \text{sin}^2(\theta) + \text{sin}(\theta) - 5 = 0$ looks a lot like a quadratic equation. If we make a substitution, it'll become even clearer. Let's replace $\text{sin}(\theta)$ with a variable, say x. So, we have:

$x = \text{sin}(\theta)$

Now, our equation transforms into:

$6x^2 + x - 5 = 0$

See? That's a quadratic equation we can work with!

2. Solve the Quadratic Equation

There are a few ways to solve quadratic equations, such as factoring, using the quadratic formula, or completing the square. Factoring is often the quickest method if it's possible. Let's try factoring this one. We're looking for two numbers that multiply to $(6)(-5) = -30$ and add up to 1. Those numbers are 6 and -5. So, we can rewrite the middle term:

$6x^2 + 6x - 5x - 5 = 0$

Now, let's factor by grouping:

$6x(x + 1) - 5(x + 1) = 0$

$(6x - 5)(x + 1) = 0$

Setting each factor equal to zero gives us the solutions for x:

$6x - 5 = 0 \implies x = \frac{5}{6}$

$x + 1 = 0 \implies x = -1$

3. Substitute Back $\text{sin}(\theta)$

Remember, we made the substitution $x = \text{sin}(\theta)$. Now, we need to substitute back to find the values of $\theta$. So, we have two equations:

$\text{sin}(\theta) = \frac{5}{6}$

$\text{sin}(\theta) = -1$

4. Solve for $\theta$

Now, we need to find the angles $\theta$ that satisfy these equations within our given range of $0^{\circ}$ to $360^{\circ}$.

Case 1: $\text{sin}(\theta) = \frac{5}{6}$

To find the angles, we'll use the inverse sine function, also known as arcsin or $\text{sin}^{-1}$.

$\theta = \text{sin}^{-1}(\frac{5}{6})$

Using a calculator, we find:

$\theta \_1 \approx 56.4^{\circ}$

However, sine is positive in both the first and second quadrants. So, we need to find the other angle in the second quadrant that has the same sine value. To do this, we subtract our reference angle from $180^{\circ}$:

$\theta \_2 = 180^{\circ} - 56.4^{\circ} \approx 123.6^{\circ}$

So, for $\text{sin}(\theta) = \frac{5}{6}$, we have two solutions: approximately $56.4^{\circ}$ and $123.6^{\circ}$.

Case 2: $\text{sin}(\theta) = -1$

This one is a bit simpler because we know the sine function equals -1 at a specific angle. Think about the unit circle: sine corresponds to the y-coordinate, and it's -1 at $270^{\circ}$.

So, $\theta = 270^{\circ}$

5. Final Solutions

We've found all the angles that satisfy the equation within the given range. Our solutions are:

$\theta \_1 \approx 56.4^{\circ}$

$\theta \_2 \approx 123.6^{\circ}$

$\theta \_3 = 270^{\circ}$

Common Mistakes to Avoid

Solving trigonometric equations can be tricky, and there are a few common pitfalls to watch out for. Let's go over some of the most frequent mistakes so you can steer clear of them.

1. Forgetting Multiple Solutions

The sine and cosine functions are periodic, which means they repeat their values over and over. This means that there are usually multiple solutions to a trigonometric equation within a given interval, like our $0^{\circ}$ to $360^{\circ}$ range. A frequent mistake is finding one solution and stopping there. Always remember to consider all possible quadrants where the trigonometric function can have the same value. We did this by finding the reference angle and then using it to find the second solution in the appropriate quadrant.

2. Incorrectly Using the Inverse Trigonometric Functions

Inverse trig functions like $\text{sin}^{-1}$, $\text{cos}^{-1}$, and $\text{tan}^{-1}$ will only give you one solution, typically within a specific range (e.g., $-90^{\circ}$ to $90^{\circ}$ for $\text{sin}^{-1}$). You need to use this solution as a reference angle and consider the periodicity and symmetry of the trigonometric functions to find all other solutions within the desired interval. This is where understanding the unit circle and the CAST rule (or ASTC rule) comes in handy.

3. Not Checking for Extraneous Solutions

When you perform operations like squaring both sides of an equation, you might introduce extraneous solutions—solutions that don't actually satisfy the original equation. It's always a good idea to plug your solutions back into the original equation to make sure they work. In our example, we didn't square both sides, so we didn't have to worry about this, but it's a good habit to have for other types of problems.

4. Calculator Errors

Calculators can be super helpful, but they can also lead to mistakes if you're not careful. Make sure your calculator is in the correct mode (degrees or radians) depending on the problem. Also, be mindful of the order of operations and use parentheses when necessary to avoid calculation errors.

5. Algebraic Mistakes

Trigonometric equations often involve algebraic manipulations, and it's easy to make a mistake with your algebra, especially when you're dealing with factoring, substitutions, or simplifying expressions. Double-check your steps and take your time to avoid errors.

Real-World Applications

So, we've solved this equation, but where does this stuff actually get used? Trigonometry and trigonometric equations are vital in many fields. Let's peek at a few examples.

1. Physics

In physics, trigonometric functions are used to describe oscillations and waves. For example, the motion of a pendulum, the behavior of sound waves, and the propagation of light can all be modeled using sine and cosine functions. Solving trigonometric equations helps physicists determine the angles and phases of these waves.

2. Engineering

Engineers use trigonometry extensively in structural design, surveying, and navigation. For instance, when designing bridges or buildings, engineers need to calculate angles and forces to ensure stability. Solving trigonometric equations is crucial for these calculations.

3. Navigation

Navigation systems, like GPS, rely heavily on trigonometry to determine positions and distances. Trigonometric functions are used to calculate the angles between satellites and receivers, which are then used to pinpoint locations on Earth.

4. Computer Graphics

In computer graphics and animation, trigonometry is used to rotate, scale, and position objects in 3D space. Trigonometric functions help create realistic movements and perspectives.

5. Music

The relationships between musical notes can be described using trigonometric functions. Sound waves are sinusoidal, and the frequencies and amplitudes of these waves determine the pitch and loudness of the notes. Trigonometry helps understand and manipulate these relationships in music production and audio engineering.

Conclusion

Alright guys, we've made it to the end! We successfully solved the trigonometric equation $6 \text{sin}^2(\theta) + \text{sin}(\theta) - 5 = 0$ for $\theta$ in the interval $0^{\circ} \leq \theta < 360^{\circ}$. We broke down the problem step-by-step, from recognizing the quadratic form to finding all possible solutions and rounding to the nearest tenth of a degree. We also discussed common mistakes to avoid and explored some of the real-world applications of trigonometry. I hope this guide has helped you build a solid understanding of how to tackle these types of equations.

Remember, practice makes perfect! Keep working on trigonometric problems, and you'll become more confident and skilled. If you have any questions or want to dive deeper into this topic, feel free to ask. Happy solving!