Solving Trigonometric Equations: Find Θ For 4sinθ - 7 = 0

Alright guys, let's dive into solving a trigonometric equation! Specifically, we're going to find all the values of θ (theta) that satisfy the equation $4 \sin θ - 7 = 0$, where θ is between 0° and 360°. And because we like precision, we'll round our answers to the nearest tenth of a degree. Buckle up, it's gonna be a trig-tastic ride!

Understanding the Problem

Before we start crunching numbers, let’s make sure we understand what the problem is asking. We have a trigonometric equation involving the sine function. Remember, sine (sin) is a ratio that relates an angle of a right triangle to the opposite side and the hypotenuse. The equation $4 \sin θ - 7 = 0$ is essentially asking us: "For what angles θ is four times the sine of that angle minus seven equal to zero?"

Also, the constraint $0^{\circ} \leq θ < 360^{\circ}$ is important. It means we're looking for solutions within one full revolution of the unit circle. Think of it like finding the angles on a clock face, but with infinitely more precision.

Key Concepts to Keep in Mind:

• Sine Function: The sine function (sin θ) gives the y-coordinate of a point on the unit circle corresponding to the angle θ.
• Unit Circle: The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the coordinate plane. It's super helpful for visualizing trigonometric functions.
• Range of Sine Function: The sine function's values always fall between -1 and 1, inclusive (i.e., $-1 \leq \sin θ \leq 1$). This is because the y-coordinate of any point on the unit circle can't be smaller than -1 or larger than 1.
• Solving Equations: Our goal is to isolate the sine function and then use the inverse sine function to find the angle θ.

Solving the Equation Step-by-Step

Now, let's get our hands dirty and solve the equation! Here’s how we'll do it:

Step 1: Isolate the Sine Function

Our first goal is to get the $\sin θ$ by itself on one side of the equation. To do this, we'll perform some basic algebraic manipulations:

1. Start with the original equation: $4 \sin θ - 7 = 0$
2. Add 7 to both sides: $4 \sin θ = 7$
3. Divide both sides by 4: $\sin θ = \frac{7}{4}$

So, we now have $\sin θ = \frac{7}{4}$, which simplifies to $\sin θ = 1.75$.

Step 2: Analyze the Result

Hold up a second! Remember what we said about the range of the sine function? It's always between -1 and 1. Our equation says that $\sin θ = 1.75$. But 1.75 is outside the allowable range for the sine function. The sine of an angle can never be greater than 1.

This is a crucial observation. It tells us that there’s something special about this equation.

Step 3: Determine if there are Solutions

Since the value we found for $\sin θ$ (1.75) is outside the range of the sine function, there are no real solutions for θ that satisfy the equation $4 \sin θ - 7 = 0$. In other words, there's no angle θ between 0° and 360° (or any angle at all, for that matter) that will make this equation true.

Why No Solution?

The reason there is no solution is fundamental to the nature of the sine function. Think back to the unit circle. The sine of an angle is represented by the y-coordinate of a point on the unit circle. Since the radius of the unit circle is 1, the y-coordinate can never be greater than 1 or less than -1.

In our equation, we ended up with $\sin θ = 1.75$. This would require a point on the unit circle to have a y-coordinate of 1.75, which is impossible. It's like trying to find a place that's both inside and outside a fence at the same time – can't be done!

Final Answer

Therefore, the final answer is that there are no solutions for θ in the range $0^{\circ} \leq θ < 360^{\circ}$ that satisfy the equation $4 \sin θ - 7 = 0$.

In summary:

• We started with the equation $4 \sin θ - 7 = 0$.
• We isolated the sine function and found $\sin θ = 1.75$.
• We recognized that 1.75 is outside the range of the sine function (-1 to 1).
• We concluded that there are no solutions for θ in the given range.

Importance of Checking the Range

This problem highlights the importance of always checking the range of trigonometric functions when solving equations. If we had blindly proceeded without considering the range of sine, we might have gotten lost trying to find a solution that simply doesn't exist. Always remember the fundamental properties of trig functions!

Let's consider another example, where we can see the sine function in action. Let's say we want to solve: $2 \sin(θ) - 1 = 0$. In that case, we get:

$2 \sin(θ) = 1$ $\sin(θ) = \frac{1}{2}$

Since 1/2 is between -1 and 1, we can proceed. What angles have a sine of 1/2? Well, $\sin(30^{\circ}) = \frac{1}{2}$. Also, $\sin(150^{\circ}) = \frac{1}{2}$. Therefore, in the range of $0^{\circ}$ to $360^{\circ}$, there are two solutions: $30^{\circ}$ and $150^{\circ}$.

Real-World Applications

While this particular equation didn't have a solution, trigonometric equations are incredibly useful in many real-world applications. They are used in:

• Physics: Describing wave motion, oscillations, and simple harmonic motion.
• Engineering: Designing structures, analyzing circuits, and modeling signals.
• Navigation: Calculating distances, bearings, and positions.
• Computer Graphics: Creating realistic animations and simulations.

The sine function, along with cosine and tangent, is a fundamental building block for understanding periodic phenomena and geometric relationships. So, mastering these concepts is essential for anyone working in these fields.

Keep Practicing!

Solving trigonometric equations can be tricky, but with practice, you'll become more comfortable with the concepts and techniques involved. Remember to always:

• Understand the problem: What are you trying to find, and what constraints are given?
• Isolate the trigonometric function: Use algebraic manipulations to get the sine, cosine, or tangent by itself.
• Check the range: Make sure the value you obtained is within the allowable range for the function.
• Use the inverse function: Apply the inverse sine, cosine, or tangent to find the angle.
• Consider all possible solutions: Remember that trigonometric functions are periodic, so there may be multiple solutions within a given range.

So, there you have it! While our initial equation had no solution, we learned valuable lessons about the range of the sine function and the importance of checking our results. Keep practicing, and you'll become a trig equation-solving master in no time!