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 x = sin 2x

Accepted Answer

Find the exact solutions of the given equation over the interval 0 to 2 pi where our equation is a square root of two sin X equals sin to X. We have four possible answers which are all different sets of angle measurements. And we also have a unit circle on the right side for helping with finding our angle. Now to solve this, let's go ahead and try to solve for X. So we first notice we have sign of two X on the right side. This is one of our known formulas which is a double angle formula where sine two X equals to two sine X cosine X. In other words, you can write this as squared of two sine X equals two sine X cosine X. Now let's go ahead and move one side to the other. Let's subtract to sin X co sine X. We end up getting the square root of two sine X minus two sin X cosine X equals zero. From here, we can factor out a sine X to get sin X multiplied by the square root of two minus two cosine of X. This equals zero. No, since we have two functions that are multiplying each other Equalling zero. We can separate these, we get sin X equals zero and a square root of two minus two cosine X equals zero. Now, we can make use of our unit circle for sin X equals zero. Equation for this one, we have sin X equals zero, which means on our unit circle, we need to find where Y is equals to zero. This occurs in two places at zero. And at pie, we can then say X equals zero, NX equals pi. Now we're going to solve our other equation for X, we have the square root of two minus two cosine X equals zero. Let's first subtract both sides by the square root of two to get negative two cosine X equals negative square roots of two. Then we can divide both sides by negative two to get cosine X equals square root of two divided by two. Now let's look at our unit circle once more. We need already the circle where X equals the square root of two divided by two. This occurs in two places. Pi divided by four and seven pi divided by four. We can then say our angles are X equals pi divided by four and seven pi divided by four. We now have four answers zero pi pi divided by four and seven pie divided by four. This is our set of answers which means the answer to our problem. This answer a Ok. I hope that we 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 x = sin 2x

Key Concepts

Trigonometric Equation Solving

Sine Function Properties and Identities

Interval Notation and Solution Sets

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