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 3θ = -1

Accepted Answer

Find the exact solutions of the equation. Consider the interval from 0 to 2 pi of the equation sine of four, the equals one. And we have four possible answers which are all angle measures of data. Just in sets. We also have a unit circle on the right side to help us solve this equation. So you have s of four the equals one. Now to solve this, let's say we want sign on the right side of our equation which means we need sun of some angle is equals to one. Well, if you look at our unit circle, we know that the sign is when our Y value equals one and this occurs at one place. This occurs at the angle pi divided by two. In other words, we can write this as sin for theta equals sine pi divided by two because we know sine pi divided by two equals one. Now we have s on both sides of the equal sign. So we can rewrite this as four theta equals pi divided by two. Now something to note is that our angle loops around the unit circle as many times as it once. So we need to add two PN to the end to account for the loop. Now let's go ahead and solve for data by dividing both sides by four. We end up getting theta equals pi divided by eight plus pi divided by two N. Now we have a potential set of angles for our answer. Let's first suggest N is equals to zero for our first loop. So we have pi divided by eight plus pi divided by two multiplied by zero to get pi divided by eight. If N equals one, we get pi divided by eight plus pi divided by two, multiplied by one, which simplifies 25 pi divided by eight. If we repeat for the other values, we will end up with the values of nine pi divided by eight N for N equals three. We get pi divided by eight plus pi divided by two multiplied by three to give us 13 pi divided by eight. If we were to shake N equals four, this value goes over to pi which is not in our domain anymore, which means we now have the answers to our problem. The equals the set pi divided by eight 55, divided by eight nine pi divided by eight and 13 pi divided by eight. If we then look at our possible answers, we can see the answer to our problem. Turns out to be answer D OK. Hope they 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 3θ = -1

Key Concepts

Solving Trigonometric Equations

Properties of the Sine Function

Interval and Multiple-Angle Considerations

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