Solving 2cosx - 1 = 0.2902 Find All Solutions In [0°, 360°)
Hey guys! Today, we're diving into a trigonometric equation and figuring out all the possible solutions within a specific range. So, grab your calculators, and let's get started! We aim to find all solutions for the equation 2cosx - 1 = 0.2902 within the interval [0°, 360°). This means we're looking for angles in degrees that, when plugged into the equation, will make it true. Trigonometric equations can seem daunting, but with a systematic approach, they become much more manageable. Our goal is to isolate the trigonometric function, in this case, cos x, and then use inverse trigonometric functions to find the angles that satisfy the equation. It’s essential to remember that trigonometric functions are periodic, meaning they repeat their values at regular intervals. This periodicity is crucial when finding all solutions within a given range. So, we'll not only find the principal solution but also any other solutions that fall within our interval of [0°, 360°). Understanding the unit circle and the properties of trigonometric functions in each quadrant will be invaluable as we solve this equation. Remember, cosine corresponds to the x-coordinate on the unit circle, so we'll be looking for angles where the x-coordinate has a specific value related to our equation. Let's break down the steps and tackle this problem together, making sure we cover all the key concepts and techniques involved in solving trigonometric equations.
Step-by-Step Solution
First, let's isolate the cosine term in our equation 2cosx - 1 = 0.2902. We need to get cos x by itself on one side of the equation. This involves a couple of basic algebraic steps. We'll start by adding 1 to both sides of the equation. This will help us move closer to isolating the cosine term. So, adding 1 to both sides gives us 2cosx = 1.2902. Now, we need to get rid of the coefficient 2 that's multiplying cos x. To do this, we'll divide both sides of the equation by 2. This will finally isolate the cos x term, giving us cosx = 0.6451. Now that we have cos x isolated, we can move on to the next step, which involves using the inverse cosine function to find the angle x. This is where our calculators come in handy! We'll be using the arccosine function (also written as cos^-1) to find the angle whose cosine is 0.6451. It's important to make sure your calculator is in degree mode since we're looking for solutions in degrees. The inverse cosine function will give us the principal value, which is the angle in the range [0°, 180°]. However, we need to consider that cosine is also positive in the fourth quadrant, so there might be another solution within our desired interval of [0°, 360°). We'll use the properties of the cosine function and the unit circle to find this second solution. Let's move on to the next part to calculate the principal value and find the other solution.
Finding the Principal Value
Okay, guys, now that we have cosx = 0.6451, it's time to find the principal value of x. This is where the inverse cosine function, or arccosine, comes into play. We'll use our calculators to compute cos^-1(0.6451). Make sure your calculator is set to degree mode so we get the answer in degrees. When you plug in cos^-1(0.6451) into your calculator, you should get approximately 49.85°. This is our principal value, which lies in the first quadrant, where cosine is positive. Now, we need to think about whether there's another angle within the interval [0°, 360°) that also satisfies our equation. Remember, cosine is positive in both the first and fourth quadrants. So, we need to find the angle in the fourth quadrant that has the same cosine value as 49.85°. To find this angle, we'll use the property that the cosine function is positive in the fourth quadrant and that angles in the fourth quadrant can be found by subtracting the reference angle from 360°. This is a crucial step in solving trigonometric equations because it helps us find all possible solutions within the given interval. We'll calculate the fourth-quadrant angle in the next part to complete our solution.
Finding the Second Solution
Alright, we've found the principal value, which is approximately 49.85°. Now, let's hunt for the second solution within the interval [0°, 360°). As we discussed, cosine is positive in both the first and fourth quadrants. So, we need to find the angle in the fourth quadrant that has the same cosine value. To do this, we'll subtract our principal value from 360°. This will give us the corresponding angle in the fourth quadrant. So, the calculation is 360° - 49.85°. Doing the math, we get approximately 310.15°. This is our second solution. Now we have two angles, 49.85° and 310.15°, both within the interval [0°, 360°), that satisfy the equation cosx = 0.6451. These are the two angles whose cosine is approximately 0.6451. It's always a good idea to double-check our solutions by plugging them back into the original equation to make sure they work. We can also visualize these angles on the unit circle to confirm that they make sense. In this case, both angles fall within the quadrants where cosine is positive, which confirms our calculations. Now that we have both solutions, let's summarize our findings and conclude our solution.
Final Answer
Okay, guys, we've reached the finish line! We set out to find all solutions to the equation 2cosx - 1 = 0.2902 within the interval [0°, 360°). We've gone through the steps, isolated the cosine term, found the principal value, and identified the second solution. Our principal value was approximately 49.85°, and the second solution in the fourth quadrant was approximately 310.15°. So, the solutions to the equation 2cosx - 1 = 0.2902 in the interval [0°, 360°) are approximately 49.85° and 310.15°. To summarize, we started by isolating cos x, then used the inverse cosine function to find the principal value. We recognized that cosine is positive in both the first and fourth quadrants, so we calculated the corresponding angle in the fourth quadrant. We ended up with two solutions within our interval. Remember, when solving trigonometric equations, it's essential to consider the periodicity of the trigonometric functions and the quadrants where they are positive or negative. This helps us find all possible solutions within the given interval. Great job, everyone! We've successfully solved this trigonometric equation. The final answer is:
x ≈ 49.85° and x ≈ 310.15°
