In Exercises 49–59, find the exact value of each expression. D...

Question

In Exercises 49–59, find the exact value of each expression. Do not use a calculator. cos (11𝜋 / 6)

Accepted Answer

Hello, today we're gonna be solving the value of the following expression. So what we are given is cosine of five pi over three. Now, the easiest way to approach this problem is to use the reference angle to find the value of our given expression. But in order to find the reference angle, we're first going to need to find the location of five pi over three on the unit circle. Now, if you're ever unsure of where a radiant is located on the unit circle, you can always convert the radiant into degrees. In order to do this, we take our given angle which is five pi over three and multiply it by the quantity 180 over pi doing. So allows us to reduce the pies in the numerator and denominator to just one. And what we are left with is five times one which is five multiplied by 100 and 80 which gives us 900. And that's going to be over three times one which gives us three, 900 divided by three, gives us 300. And what this means is that the angle phi pi over three is equivalent to 300 degrees. So the question is where is 300 degrees located on the unit circle while 300 degrees is gonna be located in the fourth quadrant of the unit circle. And now what we wanna do is you want to find the reference angle to 300 degrees while the reference angle is always located between the x axis and our given angle. And in order to get a reference angle in the fourth quadrant, we're going to take the measure of 360 degrees and subtract our given angle which in this case is 300 degrees, 360 minus 300 leaves us with 60 degrees. So that tells us that the reference angle is going to have a measure of 60 degrees. Knowing this reference angle allows us to make a right triangle for the given reference angle. And another thing this allows us to state is that cosine of five pi over three is going to be equivalent to the measure of cosine of 60 for the following right triangle that we created. So let's go ahead and draw a bigger version of this right triangle located in the fourth quadrant. So we know that the reference angle is 60 degrees. And given that this is a right triangle, we have a 90 degree angle located here. Now, the easiest way to get cosine of 60 for this right triangle is to use our special right triangles in this case here, we're going to be using a 30 60 90 degree triangle across from 30 is going to be a length of one across from 60 is going to be a length of the square root of three and across from the 90 degree angle is going to be a length of two. Now, one thing to note is that this right triangle is located in the fourth quadrant, any X value or the length of the triangle is going to be positive and any Y value or the height of this triangle is going to be negative in this quadrant. So that means the height of this triangle which is square root of three is going to be negative in the fourth quadrant. Now, what we need to do is we need to get cosine of 60 degrees for this following right triangle. Now recall that cosine of any right triangle is defined as our X value or the length of the triangle over the hypotenuse. In this case here, the length of the triangle is the X value is going to be positive one. And the Hypo news has a length of positive too. So that means that cosine of 60 is going to be 1/2 for the following right triangle. And earlier we stated that cosine of five pi over three is equivalent to cosine of 60. So what this means is that the value of cosine five pi over three is going to equal to 1/2. And with that being said, the answer to this problem is going to be c 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.

In Exercises 49–59, find the exact value of each expression. Do not use a calculator. cos (11𝜋 / 6)

Key Concepts

Unit Circle and Radian Measure

Reference Angles

Exact Values of Trigonometric Functions

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