Solving Cosine(x + Π) = 1/2: Your Math Guide

by Andrew McMorgan 45 views

Hey math whizzes and curious minds! Today, we're diving deep into the fascinating world of trigonometry to tackle a specific problem: solving the equation Cosine(x + π) = 1/2 within the interval [π/2, π]. We've also got some potential answers to consider: π/2, 2π/3, 3π/4, 5π/6. Let's break this down step-by-step, make it super clear, and figure out which of these values, if any, is our golden ticket.

Understanding the Core Equation: Cosine(x + π) = 1/2

Alright guys, let's get straight to it. Our main mission is to find the value(s) of 'x' that make the equation Cosine(x + π) = 1/2 true. But there's a catch – we're not looking for just any solution; we're specifically interested in solutions that fall within the range of π/2 to π, inclusive. This interval is crucial because it narrows down our search space. Remember, the cosine function has a cyclical nature, meaning it repeats its values. Without a defined interval, there would be infinite solutions. But with this interval, we're aiming for precision.

First off, let's talk about the cosine function itself. You guys know it, you love it (or maybe you tolerate it!). The cosine of an angle represents the x-coordinate of a point on the unit circle. The value 1/2 is a pretty common one in trigonometry, and it corresponds to specific angles. Think about your special triangles – the 30-60-90 and 45-45-90 triangles. The cosine of 60 degrees (or π/3 radians) is 1/2. Also, the cosine of 300 degrees (or 5π/3 radians) is 1/2. These are our foundational angles where the cosine value is positive 1/2.

Now, let's look at the (x + π) part. This is a phase shift. Adding π to 'x' inside the cosine function effectively shifts the graph of the cosine function to the left by π units. Alternatively, and often more helpfully for solving, we can use trigonometric identities. Remember the identity Cosine(θ + π) = -Cosine(θ)? This is a game-changer! Applying this to our equation, we get:

  • Cosine(x + π) = -Cosine(x)

So, our original equation Cosine(x + π) = 1/2 can be rewritten as:

  • -Cosine(x) = 1/2

Which simplifies to:

  • Cosine(x) = -1/2

See? We've transformed the problem into finding 'x' such that its cosine is -1/2. This is much more familiar territory for most of us.

Navigating the Interval: [π/2, π]

Now, let's talk about our playground: the interval [π/2, π]. This interval is important because it dictates where we are allowed to find our solution. Remember the unit circle?

  • π/2 radians corresponds to the point (0, 1) on the unit circle, which is straight up on the y-axis.
  • π radians corresponds to the point (-1, 0) on the unit circle, which is straight to the left on the x-axis.

So, the interval [π/2, π] covers the second quadrant of the unit circle. In the second quadrant:

  • The x-coordinates are negative.
  • The y-coordinates are positive.

Since the cosine function represents the x-coordinate on the unit circle, we are looking for angles where the x-coordinate is negative. Our target value is -1/2. This fits perfectly with the second quadrant, where cosine values are indeed negative!

We know that the reference angle for which Cosine(θ) = 1/2 is π/3. When we want Cosine(x) = -1/2, we are looking for angles in the quadrants where cosine is negative. These are the second and third quadrants.

Since our interval [π/2, π] is entirely within the second quadrant, we are looking for an angle in the second quadrant whose reference angle is π/3. The formula for an angle in the second quadrant with a reference angle 'α' is π - α.

Therefore, the angle 'x' we are looking for is π - π/3.

Let's calculate that:

  • x = π - π/3 = 3π/3 - π/3 = 2π/3

So, x = 2π/3 is our candidate solution.

Checking the Potential Values

We were given a list of possible values to consider: π/2, 2π/3, 3π/4, 5π/6. Our calculated solution is 2π/3. Let's see if it fits the bill and also briefly examine the others to solidify our understanding.

1. Checking x = 2π/3

First and foremost, let's plug our solution x = 2π/3 back into the original equation: Cosine(x + π) = 1/2.

  • Substitute x = 2π/3: Cosine(2π/3 + π)
  • Find a common denominator for the angles inside the cosine: Cosine(2π/3 + 3π/3) = Cosine(5π/3)

Now, what is the cosine of 5π/3? The angle 5π/3 is in the fourth quadrant. Its reference angle is 2π - 5π/3 = 6π/3 - 5π/3 = π/3. In the fourth quadrant, cosine is positive. So, Cosine(5π/3) = Cosine(π/3) = 1/2.

Success! Cosine(2π/3 + π) indeed equals 1/2. Now, we must ensure that x = 2π/3 lies within our specified interval [π/2, π].

  • Is π/2 ≤ 2π/3 ≤ π?
  • Let's convert to a common denominator (6): 3π/6 ≤ 4π/6 ≤ 6π/6
  • This inequality is true. 4π/6 is indeed between 3π/6 and 6π/6.

So, x = 2π/3 is a valid solution.

2. Examining Other Potential Values

Let's quickly look at the other provided values to understand why they might not be the correct solution for this specific problem, even if they are valid angles.

  • x = π/2: If x = π/2, then x + π = π/2 + π = 3π/2. Cosine(3π/2) = 0. This is not 1/2.

  • x = 3π/4: If x = 3π/4, then x + π = 3π/4 + π = 3π/4 + 4π/4 = 7π/4. The angle 7π/4 is in the fourth quadrant. Its reference angle is 2π - 7π/4 = π/4. Cosine(7π/4) = Cosine(π/4) = √2/2. This is not 1/2.

  • x = 5π/6: If x = 5π/6, then x + π = 5π/6 + π = 5π/6 + 6π/6 = 11π/6. The angle 11π/6 is in the fourth quadrant. Its reference angle is 2π - 11π/6 = π/6. Cosine(11π/6) = Cosine(π/6) = √3/2. This is not 1/2.

As you can see, only x = 2π/3 satisfies both the equation Cosine(x + π) = 1/2 and the condition that x must be within the interval [π/2, π].

The Power of Trigonometric Identities and Intervals

So, there you have it, folks! We successfully navigated the trigonometric waters to solve Cosine(x + π) = 1/2 over the interval [π/2, π]. The key takeaways here are the strategic use of trigonometric identities – specifically, Cosine(θ + π) = -Cosine(θ) – and a thorough understanding of how intervals and quadrants constrain the possible solutions.

By transforming the equation to Cosine(x) = -1/2, we made it more manageable. Then, by focusing on the second quadrant (defined by the interval [π/2, π]) where cosine values are negative, we pinpointed the required angle. The reference angle of π/3 led us directly to our solution, x = 2π/3.

This process really highlights the beauty and logic of trigonometry. It's not just about memorizing formulas; it's about understanding the relationships between angles, their trigonometric functions, and their positions on the unit circle. Mastering these concepts will make tackling even more complex trigonometric problems a breeze. Keep practicing, keep exploring, and don't be afraid to use those identities and visualize the unit circle. You've got this!

Remember, understanding the interval is just as critical as solving the core equation. If the interval had been different, say [0, 2π], we would have found two solutions for Cosine(x) = -1/2: 2π/3 (in the second quadrant) and 4π/3 (in the third quadrant). But for our specific interval [π/2, π], only 2π/3 fits the criteria. This underscores the importance of paying close attention to all constraints given in a math problem. Keep up the great work, and I'll catch you in the next mathematical adventure!