Trigonometry Solved: Find X In Cos(3x - 180°) = -√3/2

by Andrew McMorgan 54 views

Hey guys! Let's dive into a cool math problem today that's all about trigonometry. We've got an equation, cos(3x – 180°) = -√3/2, and we need to find the values of 'x' that make it true, but only within a specific range: 0 ≤ x < 180°. This means we're looking for solutions that fall between 0 degrees and just before 180 degrees. We'll be exploring how to break down this equation, find all possible angles, and then narrow them down to fit the given constraints. It's going to be a journey through the unit circle and the properties of cosine. So, grab your calculators, maybe a protractor if you're feeling nostalgic, and let's get this solved!

Understanding the Core Equation: cos(θ) = -√3/2

First off, let's tackle the heart of our problem: cos(θ) = -√3/2. This is where we need to recall our special angles in trigonometry. The cosine function represents the x-coordinate on the unit circle, and we're looking for angles where this x-coordinate is negative and specifically equal to -√3/2. We know that cos(30°) = √3/2. Since our value is negative, we need to look at angles in the second and third quadrants where the cosine is negative. The reference angle here is 30°. In the second quadrant, the angle is 180° - 30° = 150°. In the third quadrant, the angle is 180° + 30° = 210°.

So, the general solutions for cos(θ) = -√3/2 are θ = 150° + 360°n and θ = 210° + 360°n, where 'n' is any integer (positive, negative, or zero). This accounts for all possible angles where the cosine is -√3/2, as the cosine function repeats every 360 degrees. We're going to use these foundational angles to solve for 'x' in our original equation. It's like finding the base camp before we start climbing the mountain of our specific problem. Remembering these basic trigonometric values and quadrant rules is super important, guys. It's the bedrock upon which all our more complex trigonometric manipulations are built. Without a solid grasp of these fundamentals, trying to solve problems like this would be like trying to build a house without a foundation – it just won't stand!

Relating Back to Our Equation: 3x - 180°

Now, let's bring our original equation back into the picture: cos(3x – 180°) = -√3/2. Here, the 'θ' we just solved for is actually 3x - 180°. So, we can set our general solutions for 'θ' equal to 3x - 180°. This gives us two main sets of equations to work with:

  1. 3x - 180° = 150° + 360°n
  2. 3x - 180° = 210° + 360°n

Our goal is to isolate 'x' in each of these equations. Let's start with the first one. We need to add 180° to both sides to get the 3x term by itself:

3x = 150° + 180° + 360°n 3x = 330° + 360°n

Now, to find 'x', we divide everything by 3:

x = (330°/3) + (360°n/3) x = 110° + 120°n

This gives us a set of potential solutions for 'x'. Let's do the same for the second equation:

3x - 180° = 210° + 360°n

Add 180° to both sides:

3x = 210° + 180° + 360°n 3x = 390° + 360°n

Divide by 3:

x = (390°/3) + (360°n/3) x = 130° + 120°n

So now we have two general forms for our solutions: x = 110° + 120°n and x = 130° + 120°n. These represent all possible values of 'x' that satisfy the original equation, considering the periodic nature of the cosine function. It's pretty neat how we can generate an infinite number of solutions just from a few basic angles and algebraic manipulation. This step is crucial because it allows us to explore the full spectrum of possibilities before we narrow them down based on the given range for 'x'. Remember that 'n' can be any integer, so we can plug in different values of 'n' (like 0, 1, -1, 2, -2, etc.) to generate specific solutions.

Applying the Constraint: 0 ≤ x < 180°

This is where things get interesting, guys! We've found the general solutions, but the problem specifically states that 0 ≤ x < 180°. This means we only care about the values of 'x' that fall within this range. We're going to use our two general forms of solutions and plug in different integer values for 'n' to see which ones fit this requirement. Let's start with x = 110° + 120°n:

  • If n = 0: x = 110° + 120°(0) = 110°. Is 0 ≤ 110° < 180°? Yes! So, 110° is a valid solution.
  • If n = 1: x = 110° + 120°(1) = 230°. Is 0 ≤ 230° < 180°? No. This value is too large.
  • If n = -1: x = 110° + 120°(-1) = 110° - 120° = -10°. Is 0 ≤ -10° < 180°? No. This value is too small.

So, from the first general form, only x = 110° is within our specified range.

Now let's check the second general form: x = 130° + 120°n:

  • If n = 0: x = 130° + 120°(0) = 130°. Is 0 ≤ 130° < 180°? Yes! So, 130° is a valid solution.
  • If n = 1: x = 130° + 120°(1) = 250°. Is 0 ≤ 250° < 180°? No. This value is too large.
  • If n = -1: x = 130° + 120°(-1) = 130° - 120° = 10°. Is 0 ≤ 10° < 180°? Yes! So, 10° is a valid solution.

Awesome! We've found three values of 'x' that satisfy both the equation and the given range: 110°, 130°, and 10°.

Final Check and Identifying the Correct Option

We've diligently worked through the problem, and our calculated solutions are 10°, 110°, and 130°. Now, let's look at the options provided:

  • {10°, 110°, 130°}
  • {20°, 100°, 140°}
  • {30°, 330°, 390°}
  • {60°, 300°, 420°}

Our set of solutions perfectly matches the first option! It's always a good idea to double-check your work, especially when dealing with trigonometric equations and restricted domains. You can plug these values back into the original equation to confirm. For instance, let's test x = 110°:

cos(3 * 110° - 180°) = cos(330° - 180°) = cos(150°). We know cos(150°) = -√3/2. Perfect!

Let's test x = 130°:

cos(3 * 130° - 180°) = cos(390° - 180°) = cos(210°). We know cos(210°) = -√3/2. Perfect again!

And for x = 10°:

cos(3 * 10° - 180°) = cos(30° - 180°) = cos(-150°). Since cosine is an even function (cos(-θ) = cos(θ)), cos(-150°) = cos(150°) = -√3/2. Another perfect match!

It’s satisfying when everything lines up, right? This process of solving trigonometric equations involves understanding the unit circle, reference angles, quadrant rules, general solutions, and then applying specific constraints. It tests your attention to detail and your ability to work systematically. So, the correct values representing the solutions to the equation cos(3x – 180°) = -√3/2, where 0 ≤ x < 180°, are indeed 10°, 110°, and 130°. Great job following along, guys! Keep practicing these types of problems, and you'll become trigonometry wizards in no time!