In Exercises 61–86, use reference angles to find the exact value ...

Question

In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator.sin 2𝜋3

Accepted Answer

Hello, today we're gonna be evaluating the following expression by using reference angles. So what we are given is sign of three pi over four. Now, the first thing we're gonna want to do is locate three pi over four on the unit circle. Now, if you're ever unsure where a radiant is located on the unit circle, you can always convert this radiant into a degree. In order to do this, you take the given angle which in this case is three pi over four and multiply it by the value of 1 80 over pi doing. So allows you to reduce both of the pies to one. And what you are left with is three times 180 which is going to give us 540 divided by four times one, which is four and 540 divided by four is going to give us the angle of 135 degrees. So where is 135 degrees located on the unit circle? While this angle is gonna be located in the second quadrant of the unit circle. And now that we know the location of the angle, we need to go ahead and get the reference angle. The reference angle always lies between our given angle and the x axis. So this is gonna be the location of a reference angle. And since our given angle lies in quadrant two, in order to get the reference angle, we're going to be taking 100 and 80 degrees and subtracting our given angle from this value. So we're gonna be doing 100 and 80 minus 135 and that is going to give us the value of 45 degrees. So the reference angle in this case is going to have a measure of degrees. What this also says is that sine of three pi over four is equivalent to sine of degrees. So how do we solve for sin of 45 degrees? Well, since we have the location of the reference angle, we can go ahead and create a right triangle that is located in quadrant two. So let's go ahead and draw that image. So we have a right triangle located in quadrant two and the central angle is 45 degrees with a right angle of 90 degrees. Now, one way we can solve for sin of 45 degrees is to use our special right triangles. Since we have 45 90 degrees, we're going to be using the special right triangle, which is called a 45 45 90 right triangle. Across from both of the 45 degree angles is going to be the measure of one and across from the 90 degree angle is going to be the square root of two. Now, one thing to note is that this triangle is located in quadrant two, any X value or the length of the triangle in quadrant two is going to be negative and any Y value or the height of the triangle is going to be positive. So now what we need to do is we need to solve for sine of degrees for this right triangle. Now recall that sign for a right triangle is defined as our Y value over the hypotenuse. In this case here, our Y value is going to be one and our hypotenuse is going to be the square root of two. So if we plug this in to the identity of sine, sine of 45 is going to equal to one over the square root of two. But keep in mind that we cannot leave a square root in the denominator. So in order to simplify this, we're going to need to rationalize this quantity and rationalizing, we're going to be multiplying this value by the square root of two over the square root of two. This is going to leave us with sine of equal to the square root of 2/2. So we know that sine of 45 degrees is equal to the square root of 2/2. But remember that earlier, we stated that Sine of three pi over four is equivalent to sine of degrees. With that being said sign of three pi over four is going to equal to the square root of 2/2. This is going to be the solution to this problem and the answer to this problem is going to be a. So I hope this video helps you in understanding this problem. And I'll go ahead and see you all in the next video.

In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator.sin 2𝜋3

Key Concepts

Reference Angles

Unit Circle

Trigonometric Values of Common Angles

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