Solving Cosine Equations Using Inverse Trigonometry
Hey math enthusiasts! Let's dive into a cool problem involving inverse trigonometric functions. We're going to find the smallest two positive solutions for the equation $ \cos (2.5 x+4.5)=0.4231
on the interval $[0,2 \pi)$. This is a classic example of how we can use inverse trigonometric functions to solve for angles. We will break it down into simple steps, so follow along. Remember, understanding these concepts is crucial for many areas of mathematics and real-world applications. So, let's get started, and I promise, it'll be more fun than it sounds! ## Unveiling the Secrets of Inverse Trigonometric Functions **Inverse trigonometric functions** are the unsung heroes of trigonometry, allowing us to find the angle when we know the ratio of sides in a right triangle. For cosine, the inverse function (denoted as $\cos^{-1}$ or arccos) gives us the angle whose cosine is a given value. It is essential for solving trigonometric equations. These functions provide a precise way to determine angles, a crucial skill in various fields, including engineering, physics, and computer graphics. In this case, we use $\cos^{-1}$ to unravel the mystery behind the cosine equation, allowing us to solve for *x*. So, when you encounter a cosine equation where you need to find the angle, remember to call on the inverse cosine function! It's your secret weapon! Let's get our hands dirty. The equation we need to solve is $\cos (2.5x + 4.5) = 0.4231$. The first thing we need to do is isolate the argument of the cosine function, which is $2.5x + 4.5$. We do this by taking the inverse cosine of both sides. Taking the inverse cosine of 0.4231 gives us a principal value (let's call it $\theta$). Remember, the inverse cosine function gives us one angle, but there are infinitely many angles that have the same cosine value. These angles are all multiples of $2\pi$ away from our principal value. However, given the nature of our problem, we need to find the two smallest positive solutions within the interval $[0, 2\pi)$. The principal value, or the solution obtained directly from the inverse cosine, is the first solution. To find the next solution, we need to consider the properties of the cosine function on the unit circle. The cosine function is positive in the first and fourth quadrants. The inverse cosine function gives us the angle in the first quadrant. The other solution, therefore, lies in the fourth quadrant and has the same reference angle. Therefore, the two solutions can be found by manipulating the principal value. Keep in mind the periodicity of the cosine function, which repeats every $2\pi$. ### Step-by-Step Solution 1. **Isolate the Argument**: Apply the inverse cosine function to both sides of the equation: $2.5x + 4.5 = \cos^{-1}(0.4231)
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Calculate the Principal Value: Use a calculator to find the principal value of $\cos^{-1}(0.4231)$. Make sure your calculator is in radian mode:
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Solve for x: Now, we have $2.5x + 4.5 \approx 1.1337$. Subtract 4.5 from both sides:
Divide by 2.5:
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Finding the Second Solution: Cosine is positive in the first and fourth quadrants. So, besides our first answer, there is another one. The general solution for $\cos(\alpha) = \cos(\beta)$ is $\alpha = 2n\pi \pm \beta$, where $n$ is an integer. Therefore, another solution can be obtained from $2.5x + 4.5 = 2\pi - 1.1337$. Then:
So:
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Find All Possible Solutions: The general solution for $\cos (2.5x + 4.5) = 0.4231$ is:
2.5x + 4.5 = \pm 1.1337 + 2n\pi$, where *n* is any integer. Hence, $x = \frac{-4.5 \pm 1.1337 + 2n\pi}{2.5}
So, we have two families of solutions:
x_1 = \frac{-4.5 + 1.1337 + 2n\pi}{2.5}$ and $x_2 = \frac{-4.5 - 1.1337 + 2n\pi}{2.5}
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Find the Smallest Two Positive Solutions: By substituting the integer n into the above two equations, we can find all the values of x. For $x_1$, we get:
When n = 1: $x_1 = \frac{-4.5 + 1.1337 + 2\pi}{2.5} \approx 0.2597$
When n = 0: $x_1 = \frac{-4.5 + 1.1337}{2.5} \approx -1.3465$ (Not valid, as we need positive solutions)
When n = 2: $x_1 = \frac{-4.5 + 1.1337 + 4\pi}{2.5} \approx 2.7460$
For $x_2$, we get:
When n = 1: $x_2 = \frac{-4.5 - 1.1337 + 2\pi}{2.5} \approx 0.9494$
When n = 0: $x_2 = \frac{-4.5 - 1.1337}{2.5} \approx -2.2535$ (Not valid, as we need positive solutions)
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Final Answer: Therefore, the two smallest positive solutions, rounded to four decimal places, are 0.2597 and 0.9494.
Why Understanding This Matters
Solving equations involving trigonometric functions is an essential skill in mathematics. These solutions are frequently used in various applications, including physics, engineering, and computer graphics. From analyzing wave patterns to modeling periodic phenomena, understanding how to find solutions provides the foundation for further study in these areas. Also, this helps to prepare you for more complex mathematical concepts. Grasping these basics makes more advanced topics easier to understand. So, keep practicing and exploring these topics, and you'll find that it is a building block for a lot of cool stuff!
Tips for Success
To ace these problems, remember a few key tips. First, always double-check that your calculator is in radian mode. Mistakes in mode are a common pitfall. Second, understand the properties of the trigonometric functions. Knowing where each function is positive or negative, and their periodic nature, will make it easy to solve different kinds of equations. Finally, practice, practice, practice! Work through many examples to build your confidence and intuition. The more you work with these equations, the more familiar and comfortable you'll become. Remember to take your time, review the steps, and seek help when you need it. You got this, guys!
