Solving trigonometric equations Solve the following equations....

Question

Solving trigonometric equations Solve the following equations.

sin Θ cos Θ = 0, 0 ≤ Θ < 2π

Accepted Answer

Hello. Today, we're going to be solving for the angle x in the given trigonometric equation, and we're going to be writing our answers in terms of pi. So the trigonometric equation given to us is sine of 2x multiplied by cosine of 2x is equal to 0. And this is where the angle x lies within the first rotation of the unit circle between 0 and 2pi. So how can we go about solving for this trigonometric equation? Well, we're going to need the help of some trigonometric identities. Recall that the double angle formula for sin(2x) is equal to 2 multiplied by sine of x multiplied by cosine of x. If we look closely, the left hand side of the equation almost matches the right hand side of this identity. However, we are missing the coefficient of 2. So what we can go ahead and do is we can multiply both sides of the equation by 2 in order to be able to utilize this trigonometric identity. Multiplying both sides by 2 allows us to rewrite the equation as 2 multiplied by sine of 2x multiplied by cosine of 2x is equal to 0. Now we can apply the trigonometric identity and rewrite the left hand side as sine of 2 multiplied by 2x and set it equal to 0. 2 multiplied by 2x gives us the value of 4x, so we can simplify the left hand side to sine of 4x is equal to 0. Now, notice that the original domain for the angle x lies between 0 and 2 pi, but the angle that is given to us currently is 4x. So what we can do is we can edit our domain to include the angle 4x, and in order to get the angle of 4 x, we can multiply the domain by 4. This will give us the new domain, 4 x is between the values of 0 and 8 pi. So what we want to do is we want to find all the values of 4x that cause sine to equal to 0 within the region of 0 and 8 pi. Well, we can make this a little bit easier to work with by applying a substitution to both the equation and domain. We're going to allow y to equal to 4x, and if we apply this substitution, what we are left with is sine of y equal to 0 where the angle y lies within the region between 0 and 8 pi. So, we want to ask ourselves, what angles can we plug into sine that will give us the value of 0 where y is between the region of 0 and 8 pi? Well, in order to answer this, we're going to need to use a unit circle. Now, in the first notation of the unit circle, at the value of 0 we have the terminal point 1 comma 0 and at the angle pi we have the terminal point of -one comma 0. Recall that any terminal point on the unit circle is defined as cos comma sine. This is where sine is the y value of any terminal point on the unit circle. So currently, we have two solutions for y. We have y is equal to 0 and to pi. But, we want to go ahead and get all the values of y that are between 0 and 8 pi. What we can go ahead and do is we can add the period of sine to both 0 and pi and get the remaining angles that are below 8 pi Recall that the period of sine is 2 pi So, what we're going to do is we're going to add 2 pi to 0 and pi until we get angles that are less than 8 pi. Let's go ahead and separate these values. So 0+2pi will give us the value of 2pi. 2pi+another2pi will give us the value of 4pi, and 4pi+another2pi will give us the value of 6pi. Now if we were to add another 2pi to 6pi, we'll get the value of 8pi. But the domain states that we do not want to include the value of 8pi, so so we're going to stop at the value of 6pi. Now, pi+2pi will give us the value of 3pi. 3pi+another2pi will give us the value of 5pi. And 5 pi plus 7 pi will give us the value I'm sorry. 5 pi plus 2 pi will give us the value of 7 pi. Now if we were to add another value of 2 pi to 7 pi, we'll get the value of 9pi which lies outside our domain between 0 and 8pi. So we're not going to include that value and stop at 7pi. So in total, we have 8 solutions for y. Y is equal to 0, 2pi, 4pi, 6pi, and y is equal to pi, 3pi, 5pi, and 7pi. These 8 angles will satisfy the equation sinY=zero within the domain of 0 and 8 pi, but keep in mind that we are not solving for y, we want to solve for x. So if we reapply our substitution of y is equal to 4x, we can rewrite the solutions as 4x is equal to 0, pi, 2pi, 3pi, 4pi, 5pi, 6pi, and 7pi. And now if we want to solve for x, we're just going to divide every solution within this equation by 4. This will give us x is equal to 0 divided by 4, which is 0, pi divided by 4, which will give us pi over 4, 2pi divided by 4, which will simplify, 2pi divided by 2, 3pi divided by 4, which will leave us with 3pi divided by 4, 4pi divided by 4, which will leave us with just pi, 5pi divided by 4, which cannot be simplified, so we have 5pi divided by 4, 6pivivived by 4, which will simplify to 3pitwo, and 7pivivived by 4, which cannot be simplified, so we have 7pivived by 4. These 8 angles are going to be the values of x that satisfy the original 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.

Solving trigonometric equations Solve the following equations.

sin Θ cos Θ = 0, 0 ≤ Θ < 2π

Key Concepts

Trigonometric Functions

Zero Product Property

Interval Notation

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