In Exercises 85–96, use a calculator to solve each equation, corr...

Question

In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅).

2
cos x = ﹣ ------
                    5

Accepted Answer

Hello, everyone. We are asked to solve the trigonometric equation on the interval of to 2 pi including zero with the help of a calculator, we will round our final answer to four decimal places. The equation we are given is the cosine of X equals 4/5. We have four answer choices that are all very, very close decimals. They vary in the last decimal place. So to begin with thinking about the cosine of X equals 4/5 I noticed that here 4/5 is positive. So this means I need to think about what quadrants have positive cosines and that would be quadrant one and quadrant four. So if I were to just sketch a quick coordinate plane quadrant one and quadrant four are gonna be the two quadrants we are working with to find this angle. Notice in our answer choices, each choice has two values. This means that in the end, we should have two angles and we will one in quadrant 11 in quadrant four. First, we'll find the one that's in quadrant one. And we will do that by doing the arc cosine of 4/5 equal to X. When you plug this into a calculator. Be careful to make sure your calculator is in radiant mode or you will get a completely different answer. So when this is placed in a calculator, X would equal 0.643501. And since we want to round to four decimal places, it would be 0. this is one of our angle measures. So that's the one in quadrant one. And I just drew an arbitrary angle. So I can label it as 0.6435 in quadrant one. Next, thinking about quadrant four, if it has the same value as the one in quadrant one, this means that it has spacing from our x axis. That is also 0.6435 for distance. But that's not gonna be the angle we wanna find, we wanna find the angle that's in standard position. So the one that starts at the positive X axis and goes all the way around till we're in quadrant four. Recalling that the full rotation of a circle around our quadrants is two pi I can find this angle by doing two pi minus 0.6435. And when I do so I get 5.6397. So that would be the measure of my larger angle here. The one that almost makes a full rotation and it is rounded to four decimal places. So now looking for matches. I see that 0.6435 would be answer choice B or D and then 5.6397 makes it answer choice. D have a nice day.

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In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). 2 cos x = ﹣ ------ 5