Solve the following equations.

Question

$\cos3x=\sin3x,0\leq{x}\lt2\pi$

Accepted Answer

Hello. Today we're going to be solving for the angle of a in the following trigonometric equation, then we'll be writing our answer in terms of pi. So the trigonometric equation given to us is sine of 2a equal to cosine of 2a and this is where the angle a lies between the values of 0 and 2pi not including 2pi. So, how can we go about solving for a in the following equation? Well, we can first simplify the equation by dividing both sides of the equation by cosine of 2a. Doing this will allow us to rewrite the equation as sine of 2a divided by cosine of 2a and this quantity will equal to 1. Next, we're going to use a trigonometric identity to help us simplify the left hand side of the equation. Recall that tan is equal to sine of x divided by cosine of x. If we were to apply this trigonometric identity on the left hand side, we can rewrite the equation as tangent of 2a is equal to 1. Now, one thing to note is that the original domain for A stated that A must lie within the first rotation of the unit circle. However, this is for an individual angle of A. In the current equation, we have a double angle of a, so we're going to need to edit this domain in order to have 2a within the domain. In order to do this, we can multiply the domain by 2 and this will allow us to rewrite the domain of a as 0 less than or equal to 2 a less than 4 pi. So what this means is what angles of a can we plug into tangent of 2 a to get the value of 1 and the angle 2 a must be between 0 and 4 pi? Well, in order to answer this question, we're going to need to use a unit circle. But before we use the unit circle, let's go ahead and apply a substitution to both the new domain and the current equation. We're going to allow y to equal to 2 a. This will allow us to rewrite the equation and the domain as tangent of y is equal to 1, and this is where 0 is less than or equal to y which is less than 4 pi. So let's go ahead and create the unit circle. Now we need to ask ourselves where is tangent of y equal to 1 on the unit circle or what angles of y can we plug into tangent to give us the value of 1? Well, there are 2 total angles on the unit circle. The first angle is at pi divided by 4 and the second angle is at 5pi divided by 4. So currently we have 2 solutions for y. We have y is equal to pi divided by 4 and y is equal to 5pi divided by 4. Now the domain states that we want all the values of y that are between 0 and 4pi not including 4pi. So far we have the first two angles between 0 and 2pi So how can we get the remaining angles in order to get angles all the way up to 4pi? Well, recall that the period for tangent is pi. So what we can do is we can add pi to pi over 4 which will give us 5pi over 4. Then we can continue adding pi to the current angle until we get an angle that is less than 4pi or does not exceed 4pi. So if we were to add pi to 5pi divided by 4, the next angle we will get is 9pi divided by 4. And if we add another angle of pi or another period of pi to 9pi divided by 4, the next angle we get is 13pi divided by 4. We'll go ahead and end right there because the angle 13pi divided by 4 is slightly less than 4pi and we don't want to exceed or equal the angle 4pi. So on the domain from 0 to 4pi, there are tol 4 total solutions for y. Y can equal to pi divided by 4, 5 pi divided by 4, 9 pi divided by 4, and 13 pi divided by 4. Now keep in mind that we wanna solve for the angle a, so let's go ahead and resubstitute in the value for 2a for y. This will allow us to rewrite the solutions as 2a is equal to pi divided by 4, 5pi divided by 4, 9pi divided by 4, and 13pifour. Now if we want to solve for a, what we can do is we can divide every solution in the current equation by 2. This will allow us to rewrite the left hand side as just a and pi divided by 4 divided by 2 will give us the angle pi divided by 8. 5pi divided by 4 divided by 2 will give us 5pi divided by 8. 9pi divided by 4 divided by 2 will give us 9pi divided by 8 and 13pi divided by 4 divided by 2 will give us 13pi divided by 8. These are going to be the 4 solutions for the angle a that satisfy the original equation. And with that being said, the answer to this problem is going to be b. 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.

Solve the following equations.
$\cos\)3x=\(\sin\)3x,0\(\leq{x}\]\lt\)2\(\pi$

Key Concepts

Trigonometric Identities

Solving Trigonometric Equations

Periodic Functions

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