Solve each equation (x in radians and θ in degrees) for all ex...

Question

Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.
sin x (3 sin x - 1) = 1

sin x (3 sin x - 1) = 1

Accepted Answer

Hello, everyone. We are asked to find the solutions of the given equation. We want to express our answer in radiance using the least possible nonne angle measure. And we're gonna round our answer to four decibel places if necessary. So we are given sine X multiplied by four, sin X plus three in parentheses equals five. Our answer choices are a, the set 0.9344 plus two N pi 2.2072 plus two N pi where N is any integer B, the set 0.9344 plus two N pi where N is any integer, C the set 2.2072 plus two N pi where N is any integer, D no solution. So I'm going to rewrite the original equation. So I have the sine of X multiplied by the parentheses. Four sx plus three, close parentheses equals five. So the first thing I thought was, oh, does this equal zero? And I can just use the zero product theorem. Unfortunately, it does not. So I'm gonna factor this probably. But first I need to multiply. So I'm distributing sin X into the parentheses. So sx multiplied by four sine X will give me four sine squared X sin X multiplied by three, give me a positive three sine X and it still equals five. But I'm gonna subtract five from both sides. So I can say minus five equals zero because this way, if it equals zero, at least it's factor or I can see what else needs to be done. Unfortunately, for us, this is not factor. So we are gonna use the Pythagorean theorem. Um Recall that the Pythagorean theorem typically says X but we're gonna use it as sin X equals the opposite of B plus or minus the square root of B squared minus four AC all over two A. So in our example, A is four, B is three C is negative five. So I'm gonna move over a little bit to the right and work on that. So sine X will equal the opposite of B. So negative three plus or minus the square root of three squared minus four times A which is also four times C which is negative five, all of it's divided by two multiplied by four because that's a, so let's clean this up. So negative three plus or minus the square root of three squared which is nine, negative four multiplied by four, multiplied by negative five will give me a positive 80. And in my denominator, I will get eight. So what is under my radical? I have negative three plus or minus the square root of 89 divided by eight. So I know that's what the sine of X equals from there. I don't see anything else that reduces. So we're gonna use this to make two equations to solve. The first one will be that the sine of X equals negative three plus radical 89 divided by eight. And the other is that the sine of X equals negative three minus radical 89 divided by eight. So I'm gonna use my inverse sign. So the inverse sign of negative three plus radical 89 divided by eight equals X. And that's what I'm gonna put in a calculator. Make sure that you are in radiant mode because that's what we're looking for here. And when I do this, I get the X equals 0.9344. So that's one of my X values. But since we want it for every possible solution, this needs to be more of an expression. So X is gonna equal 0.9344 plus an unknown amount of rotations. So it would be two pi N and this is where N is any integer because we don't know if it's one rotation or a million rotations. So this is part of our answer. So let's finish the other part. Going back up. I have the sine of X equals negative three minus radical 89 divided by eight. So again, I'm going to use the inverse sign and they'll be of negative three minus the square root of 89 divided by eight, that'll equal our X value. And again, using a calculator, X would be 2.2072. So again, I need to write this as an expression because we don't know how many rotations we're taking into account. So X will equal 2.2072 plus two pi N to account for our rotations. And that's gonna be where N is any integer. So now that we have two answers that should represent all of the possible answers. Let's see what I have as options. Well, this appears to match a answer choice. A, the only difference is instead of writing two pi N, they wrote two N pi but that means the same thing. So answer choice A is our answer. Have a nice day.

Key Concepts

Solving Trigonometric Equations

Radians and Degrees Conversion

Using the Unit Circle and Reference Angles

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