Solve each equation in x over the interval [0, 2π) and each eq...

Question

Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.

sin (θ/2) = csc (θ/2)

Accepted Answer

Find the exact solution of the given equation over the interval from 0 to 360 degrees side of the divided by three equals cos of data divided by three with four possible answers, 90 degrees, 2, 70 degrees, 9270 or no solution. And on the right side, we have a unit circle to help fight for insert. Let's first rewrite this with S A divided by three equals cos theta divided by three. Now, one thing we know is that cos seeking is the reciprocal function one divided by S which means we can rewrite this as sign they divided by three equals one divided by S data divided by three. Now let's multiply both sides. We'll end up kidding sign they divided by three squared is equals to one if we multiply both sides by sign data divided by three. And now we can subtract one from both sides to get sine beta divided by three squared minus one equals zero. Now, here we can make use of an algebra formula which is the difference of two squares, which tells us if we have something A squared minus B squared. The result is the same as A plus P multiplied by A minus P. In our case, we gets data divided by three plus one multiplied by sine beta divided by three might have sweat above equals zero. Now, since we have a product of the two expressions equaling zero, we can separate these side, they are divided by three plus one equals zero. And sine data divided by three minus one equals zero, giving us two expressions sine data divided by three equals one or negative one. Now, here we can make use of the unit circle and figure out where we have a sign equaling one and negative one. If we look at our circle, we see it, the sine or the Y value equals one at pi divided by two. And on the bottom it equals negative one at three pi divided by two, we can now write these instead theta divided by three equals pi divided by two F theta divided by three equals three pi divided by two. Now another thing to note is that this is in degrees. So we have to change our radiant angles to degrees by saying theta divided by three equals 90 theta divided by three equals 270. Now, we know something about the sign function is that the side function loops around the unit circle every time it goes 360 degrees. In other words, if we move 360 degrees from our current angle, we would end up at the same angle. So for every one of our angles, we are going to add 360 degrees multiplied by N for the number of loops. Now we have set up our general solutions and we can go ahead and solve your data. We multiply both sides by three and get theta equals 270 plus in 80 multiplied by N and theta equals 270 multiplied by three, which gives us 8, 10 plus 1080 multiplied by it. There's something we notice here about our second equation. We have 810 degrees which we know that value is greater than our end of our interval. 360. Because the number is larger, we will not use that equation. Which means we will just look at our first equation. If we use our first index and equals zero, we get 270 plus 1080 multiplied by zero, which is just 270 degrees. Any larger index will put us over 360 which means our answer to 70 degrees just corresponds with answer C OK. I hope to help you solve the problem. Thank you for watching. Goodbye.

Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.
sin (θ/2) = csc (θ/2)

Key Concepts

Trigonometric Functions and Their Definitions

Solving Trigonometric Equations

Interval and Angle Measurement Conventions

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