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

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.

2 - sin 2θ = 4 sin 2θ

2 - sin 2θ = 4 sin 2θ

Accepted Answer

Hello, today we are going to be finding all of the solutions to the given trigonometric equation. We are going to express our answer in degrees using the least possible nonne angle measure. We will also be rounding our answer to the nearest 10th if necessary. So let's go ahead and take a look at the given equation. We are given sign of two, theta plus three is equal to five of sine of two theta. In order to solve for theta, we will need to simplify the equation first since we have sin of two theta on both the left and the right hand side, what we can do is we can subtract five of sin of two theta to both sides of the equation. This will allow us to rewrite the equation as negative four of sine of two. Theta plus three is equal to zero. Next, what we want to do is we want to isolate sine of two theta on the left hand side. In order to do this, we are going to subtract three to both sides of the equation that will leave us with negative four of sine of two, theta is equal to negative three. Next, what we are going to do is we are going to divide both sides of the equation by negative four. That will leave us with sine of two. Theta is equal to three divided by four. Now, what we are going to do is we are going to apply a substitution to the angle to theta. If we allow Y to equal to two theta, we can rewrite the equation to be sign of Y is equal to three divided by four. Next, what we want to do is we want to isolate Y from the sine function. In order to do this, we will be taking the sine inverse of both sides of the equation. And that will allow us to write Y is equal to the sine inverse of three divided by four. Now, what we are going to do is we are going to evaluate this quantity. So in order to evaluate this quantity, we will need the help of a unit circle and we will need a calculator. Now, before we plug in the value into a calculator, it's important to know that the domain of sign inverse is from negative 1 to 1 or is from negative pi divided by two to pi divided by two. What this means is that the domain of sine inverse is the entire right half of the unit circle. This is important to know because when you plug in sine inverse of three divided by four into a calculator. The result is going to be an angle that is located within this domain. So if we plug ins inverse of three divided by four into a calculator, what we are going to end up returning is 48.59 degrees. So we have the angle of 48 0.59 degrees located in the first quadrant of the unit circle. Now, since we have a positive value for sign, there must be another equivalent value that is positive to this angle, that the only other quadrant that is positive for sign is going to be quadrant two. The other angle for the solution must be an angle within quadrant two. That also has a reference angle of 48.59 degrees. In order to solve for the second solution, we are going to take 180 degrees and subtract 48.59 degrees from this value 180 minus 48.59 degrees will give us an angle of 131.41 degrees. So in total, we have two solutions for why the first solution is going to be 48.59 degrees. And the second solution will be 131.41 degrees. So we can write those solutions down as Y is equal to 48.59 and Y is equal to 131.41. Now notice that the original problem does not tell us that we have a restriction on the domain for any of the solutions. What this means is that we have the solutions of 48.59 and 131.41. But we must, must also include all the co terminal angles to both of these solutions. In order to include all of the co terminal angles, we must think about the period for sign, the period for sign is given to us as to pi but all the co terminal angles are any angles that we can add 360 degrees to these values or subtract any amount of 360 degrees. So we're going to allow two pi to equal to 360 degrees. And we're going to multiply this value by an integer N. So what we are going to do is we are going to add the value 360 N to both of our solutions that will allow us to write the, write the solutions including all the co terminal angles. So we have Y is equal to 48.59 plus 360 N. We will also have Y is equal to 131.41 plus 360 N. These are going to be the two solutions for why? But keep in mind is that we want we want to solve for theta originally. So now what we are going to do is we are going to res substitute two theta for why? So we now have two, theta is equal to 48.59 plus 360 N and two, theta is equal to 131.41 plus 360 N. In order to solve for theta, in both of these equations, we will need to divide both of the equations by two. That will leave us with theta is equal to 24.3 plus 180 N and theta is equal to 65.7 plus 180 N. These are going to be the two solutions for the angle theta in the given trigonometric equation. And with that being said, the answer to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Key Concepts

Solving Trigonometric Equations

Double-Angle Identities

Angle Measurement and Conversion

Watch next