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. sec 7𝜋 / 4

Accepted Answer

Hello, today we're gonna be determining the following value of the given expression. So what we are given is second of 11 pie over six. Now, one way to approach this problem is to use reference angles. And in order to get the reference angle, we're going to need to find the location of 11 pi over six on the unit circle. If you're ever unsure of where 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 11 pi over six and multiply it by the quantity 180 over pi doing this allows us to reduce the pies in the numerator and denominator to just one. And what we are left with is 11 times 180 which is going to give us the value of 1980. And then the denominator we have six times one, which is going to give us six, 1980 divided by six is going to give us the value of 330. So what this means is that the angle 11 pi over six is equivalent to 330 degrees. So now the question is where is 330 degrees located on the unit circle while 330 degrees is gonna be located in the fourth quadrant of the unit circle. And now that we have the location of 330 degrees, we need to calculate its reference angle. The reference angle is located between the x axis and our given angle. And in order to get a reference angle in the fourth quadrant, we take the angle measure of 360 degrees and subtract our given angle which is 330 degrees, 360 minus 330 is going to give us an angle of 30 degrees. So what this means is that the reference angle is going to have a measure of degrees. Now that we have the reference angle, we can use this to make a right triangle in the fourth quadrant. This also allows us to say that sin of 11 pi over six is going to be equivalent to second of degrees for the following white right triangle we created. So let's go ahead and draw a bigger version of this right triangle. So we have a right triangle that's located in the fourth quadrant. We know that the reference angle is 30 degrees and we have a degree angle located here. Now, in order to solve for seeking of 30 we can go ahead and use the special right triangle, which is a 30 60 90 degree triangle. So we have a 60 degree angle located here across from degrees is going to be a length of one across from 60 degrees is going to be a length of square root of three and across from 90 degrees is going to be a length of two. Now, one thing to note is that this right triangle is located in the fourth quadrant that means any X value or the length of the triangle is going to be positive in this region and any Y value or the height of the triangle is going to be negative in this region. So what this means is that the height of this right triangle is going to be negative one. So now that we have the side links, we can use this to solve for second of 30 degrees of this following right triangle. Now recall that second is the inverse of cosine and that is defined as our hypotenuse over the X value of the right triangle. So in this case here, the X value of the right triangle is square root of three and the hypotenuse is two. So that means that seeking of 30 degrees is going to be two over the square root of three. Now, one thing to note is that we cannot leave a radical in the denominator, we're going to need to rationalize this quantity. So in order to rationalize we're going to multiply the numerator and denominator by the square root of three. And doing this allows us to simplify the quantity as two times the square root of 3/3. So what this means is that second of 30 degrees of the following right triangle is equal to two square root 3/3. And earlier, we stated that seeking of 30 degrees is equivalent to seeking of pi over six. What this means is that the value of sequence 11 pi over six is going to equal to two square root 3/3. This is going to be the value for the given expression. 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. sec 7𝜋 / 4

Key Concepts

Understanding the Secant Function

Reference Angles and Unit Circle Values

Sign of Trigonometric Functions in Quadrants

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