In Exercises 23–34, find the exact value of each of the remain...

Question

In Exercises 23–34, find the exact value of each of the remaining trigonometric functions of θ. cos θ = 8/17, 270° < θ < 360°

Accepted Answer

Hello everybody. I have everything. All right. Today, today we're going to be talking about this question that states determine the value of the five other trigonometric ratios where they tell us that cosine of theta is equal to divided by 65. And our theta angle lies between 270 degrees and degrees. So 270 degrees is less than theta and theta is less than degrees. So when they tell us that cosine of theta is equal to 16 divided by they tell us where our data angle lies. We have to think about how we're going to graph this. Now, of course, in the is positive in two specific quadrants in a graph in a, in a plot. So cosine theta is positive in quadrants and I'm focusing on the positive because the cosine data that they gave us was positive. So we are positive in quadrants one and four. And when you think about cosine the, think about the X axis, so you have to think about where is the X axis positive? And that is to the right side of the Y axis and the quadrants to the right side of the Y axis are quadrants one and four. Now, we look at what they, where they told us the was so 270 degrees is less than theta is less than 360 degrees. Now, this means that theta is in quadrant four. So we see that quadrant four is shared between the data angle and the cosine data they gave us. So we can confirm that we are indeed in quadrant four. So before we move on, let's talk about Soka, which is an important rule in trigonometric ratios. Soto being S O H dash C A H dash T O A and I can explain it a bit more as we go specifically, I'm going to use the C A H abbreviation for my example, the C, so the first letter in each three letter abbreviation stands for either sign cosine or tangent. So in our case, we're dealing with cosine, so we'll use C C A H. So the C stands for cosine data. So we'll have that cosine theta equals and then the second two letters. So we have a, which can stand for adjacent O can stand for opposite and H can stand for hypotenuse. So if we have A H we have adjacent and hypotenuse, and what we want to do is divide the first letter by the second letter. So we have C A H which means C cosine theta is equal to a adjacent divided by the second letter H hypos. And now they told us that this is equal to 16, divided by 65. So the adjacent side of our theta angle has a measurement of 16 and the hypotenuse has a measurement of 65. So we can use that information to create a small graph where we draw a Y axis and an X axis with an emphasis on quit number four. So since that's where we will be, and we draw a line coming out of the origin and into quadrant four, and then we'll connect that to the X axis making a right triangle angle label are the angles and then we can label each side. So we said that the adjacent side was 16 and the hypotenuse was 65 and we label the other angle the opposite or the opposite side as B. And why do I label as B? Well? Because when we have a right triangle where we're trying to find the measurements of the sides, we use Pythagorean theorem. And Python and theorem says that A squared plus B squared equals C squared where the A and the B are the legs of the right triangle. And the C is the hypotenuse. So we have 16 squared plus B squared equals 65 squared or that plus B squared is equal to 4000, 225. And Now, if I move the 256 over to the right hand side, it, we'll have that B squared is equal to 3969. And if I take the square to both sides, I will be left with B is equal to 63. And now before I move on to, since now I have all the sides of my triangle, I'm able to calculate all the trigonometric ratios. But before I move on to that, let's just make a note of quadrant four, which is where we will be, let's note that quadrant four in quadri four, only cosine of data and S data are positive. So once again, remember when you're thinking of cosine, think of the X axis. So where is the X axis positive? And when you're thinking of sine think of the Y axis, so the y axis is positive in quadrant one and two because it's above the X axis, but it's negative in quadrants three and four because it's below the X axis. So let's say we're doing sign of the, we know sine will be negative since we're in quadrant four. So sine of the will be negative. And if we look at so we'll have sine or S is sine is equal to O which is hypo or I mean, o which is opposite, divided by H which is hypotenuse. So we have opposite was, which was the B. So we have negative 63 divided by the hypotenuse which was 65. So sine theta is equal to negative 63 divided by 65. And then they told us cosign data. So we can move on to tangent data, tangent data. We look at the, we have T O A tangent is equal to opposite, divided by the A which is adjacent. So opposite is B 63 divided by adjacent, which is 16. And remember this will also be negative because the only things that will pos be positive are cosine and C can. So then we can move on to coy data. Now cos theta is equal to one divided by sine theta or the reciprocal of sine theta. So we just take the sin theta that we already calculated and we flip it. So we have negative 65 divided by 63 now and we can move on to C theta C theta is equal to one divided by cosine theta. So once again, the reciprocal of the cosine and they told us the cosine. So we'll have 65 divided by 16 and this will be positive. Now we move on to cotangent theta lastly and co tangent is one divided by tangent theta. So the reciprocal of the tangent that we calculated. So that would be negative 16 divided by 63. And just like that, now we have all six trigonometric ratios where they gave us one of them being cosine and using our graph and some simple calculations, we were able to calculate the other five. So I hope that was helpful. And until next time.

In Exercises 23–34, find the exact value of each of the remaining trigonometric functions of θ. cos θ = 8/17, 270° < θ < 360°

Key Concepts

Trigonometric Functions and Their Relationships

Quadrant Sign Rules

Using the Pythagorean Identity to Find Sine

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