Solve the following equations.

{Use of Tech} ...

Question

Solve the following equations.

{Use of Tech} $\sin2 heta=\frac15,0\lt{ heta}\lt{\frac{\pi}{2}}$

Accepted Answer

Hello, today we're going to be determining the value of theta in the following trigonometric equation. So we are told that cosine of 2 theta is equal to 1 divided by 2, and we are told that the values of theta must be between the region of 0 and pi on the unit circle. So, notice that in the trigonometric equation, we are given the angle 2 theta. We're going to need to edit our domain in order to include this double angle. In order to edit the domain to include 2 theta, we're going to multiply the entire inequality by 2 and this will allow us to rewrite the domain as 0 less than 2 theta less than 2 pi. So, we have cosine of 2 theta is equal to 1 divided by 2 and this is where 2 theta is between the region of 0 and 2pi. In order to solve this trigonometric equation, let's go ahead and apply a substitution to both the equation and the domain. We're going to allow y to equal 2theta. So we can rewrite the equation as cosine of y is equal to 1 divided by 2, and this is where the angle y must be between the region of 0 and 2 pi, or in other words, we only want the values of y within the first rotation of the unit circle. Now in order to answer this equation, we're going to need to use a unit circle. Now what this equation is asking us is what angles of y can we plug into cosine that will give us the values of 1 divided by 2? Now recall that every terminal point on the unit circle is defined as cosine comma sine. This is where cosine is the x value of every terminal point. Or in other words, we can think of the equation as what angles have a terminal point that have the value 1 divided by 2 in the x position. While the first angle can be found in the first quadrant at pi divided by 3. The terminal point at this angle is 1 divided by 2 comma square root 3 divided by 2. Since we have 1 divided by 2 in the x position, this will satisfy the equation. The next angle can be found in the 4th quadrant at the angle 5pi divided by 3. At this terminal point, we have the value 1 divided by 2 comma negative square root 3 divided by 2. Again, we have 1 divided by 2 in the x position, so this angle will satisfy the equation. So, what this means is that there are 2 solutions to y in cos y is equal to 1 divided by 2. Y is equal to the angle pi divided by 3 and 5 pi divided by 3. Now recall that we want to solve for the angles theta. So what we're going to do is we're going to resubstitute 2 theta for y. Doing this will allow us to rewrite the solutions as 2 theta is equal to pi divided by 3 comma 5 pi divided by 3. And in order to solve for theta, we're just going to need to divide every term by 2. This will leave us with theta is equal to pi divided by 3 divided by 2 which will give us pi divided by 6, and 5pi divided by 3 divided by 2 which will leave us with 5pi divided by 6. These two solutions to theta will satisfy the original trigonometric equation, and these angles of theta lie between the region of 0 and pi. And with that being said, the answer to this problem is going to be D. 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.

{Use of Tech} $\sin2\theta=\frac15,0\lt{\theta}\lt{\frac{\pi}{2}}$

Key Concepts

Trigonometric Functions

Double Angle Formulas

Interval Notation

Watch next