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Solve the equation on the interval 0 < 0 < 21V3 sin (40) =OA: 00B. 0, 4'II 21 7r Tr 131 51 191 0c. 12' 6 ' 3 ' 12' 6 ' 12 3 ...

Question

Solve the equation on the interval 0 < 0 < 21V3 sin (40) =OA: 00B. 0, 4'II 21 7r Tr 131 51 191 0c. 12' 6 ' 3 ' 12' 6 ' 12 3 ' 12510D.

Solve the equation on the interval 0 < 0 < 21 V3 sin (40) = OA: 0 0B. 0, 4'I I 21 7r Tr 131 51 191 0c. 12' 6 ' 3 ' 12' 6 ' 12 3 ' 12 51 0D.



Answers

Find all solutions on the interval $[0,2 \pi)$. $$ 2 \sin ^{2} w+3 \sin w+1=0 $$

Have to simplify this equation. The question is four scientist er plus three multiplayer by square atop Terry is equal to square luxury. First we subtract three square, a doctorate from both sides, figured four. ST Rita is Equality Square atop Terry minus Terry squared off. Three. This is equal to minus toe the square of tree now really, where both sides by four we get 78 a. Is equal to minus toe multiplied by squirrel looked really worried by four. This is equal to minus escargot. Talked really right by two. And we know that scientists is negative intern and forth Gordon, so this will be able to sign off by plus by anywhere by Terry, is required to sign off two by minus by divided by three. This is equal to sign off for by Divided by Terry is equivalent to sign off. Five by divided by three. So Trita is equal toe for by divided by Terry, and Toyota is equal to five by divided by Terry. It means poor by beware. Valtteri Coma five. By divided by Terry Is the solution set off this prison

We have to solve. So sign Esquire Tater minus five. Scientific data blustery is equal to zero. This can berating else to sign Esquire Tater minus two. Scientific data minus study. A scientist a plus study is equal to zero where Tater belongs to closure and travel. Zero toe open interval to buy. Now this if in fact raised this then this candidate and head signed to term minus one multiplied by So scientists minus 30 is equal to zero. The sword isn't over. This is either scientific is a quarto one or scientist is required to three divided by two. But we know that maximum value off scientific data is one. So there is no sort Listen from here and from here Bigot Tita is equal toe by divide by two. This on this angle scientist Isaac for one. So singleton pie, divided by two, is the only soldiers and off this prison

So to solve this equation for theta, let's begin by applying a some to product formula. Formula seven from the section will help us out here. So let's call forth IATA Alfa and six Data Beta. Then we know that sign. Alfa minus sign beta is equal to two time sign of Alfa minus beta. So forth. Ada minus six Data. Oh, too. Times co sign of Alfa plus beta. So in this case, fourth data less six data over to not equal zero. So then, if we divide by two on both sides, the two bets eliminated here and zero divided by two is still zero, so we'll still be equal to zero. Then let's simplify the expressions inside of signing co sign so forth. Beta minus six. Data is negative. Two fada to dividing by two that is naked of Fada. And here adding together for data and six data. We get 10 data writing by two. So we get co sign of five feet. All right, so now we have to products that multiplied together equals zero so we can apply the zero product property. We have two possibilities. Either sign of negative Fada equals zero. Oh, co sign of fact data equals era, and suddenly they can both be zero as well. So now we're going to solve each of these separate equations, and we want to keep getting solutions from 0 to 2 pi. So if sign of negative data equals zero, the naked of data must be equal to negative two pi or negative pie or zero or pie or two pi All these places where sign of these ankles equals zero. So then dividing by negative one, we get faded, equals, do you pi pi zero negative pie. Negative two pi it, sire. So we only want to take the values that are in between zero and two pi and we're not going to be including two pi. So the values that fit that rpai and zero so we have fada equals zero or data equals pi. So these are our first solutions. We're not done, though, as we make more solutions from the second part, co sign five beta equals zero. So let's all that now. So we need to list all the places where we have a co sign value of zero. So this occurs at pi over two three pi over two five high over to etcetera. So all these places, these odd intervals of high over to So let's divide by five at then will list all of the solutions until we got to two pi. So dividing by five we get Fada equals pi Over 10 three pie over 10 Fifi over 10 which is high over to seven pi over 10 nine a pie over 10 11 pi over 10. 13 pi over 10. 15 pi over 10 on. We could reduce that. She's three pi over two 17 pi over 10 19 pie over 10 and will stop there as anymore And we would get above two pi so we have to stop here. So this is our list of solutions. And so Tikrit a complete list will add in zero and pie from earlier. So also zero and high because they want to have the complete list of all solutions from both pieces of this equation. So we have this very large box of all of these solutions. So each of the is, well, Syria. That's the solution to this equation. So this here is our final answer

Given Sign Acts equals one half. We're going to use the unit circle to help us see where that is true. So sign X is in our Y position, so we want that second value to be one half, Sin X is one half. We find that at 30 degrees, we get that one half in the second value And 150°.. There's one half in that second value, so sign axes 1/2 at 30 degrees And 150°..


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