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

Question

In Exercises 23–34, find the exact value of each of the remaining trigonometric functions of θ.cos θ = -3/5, θ in quadrant III

Accepted Answer

Hello everybody, everything. All right. Today, today we're going to bring out this question that states determine the value of the five other trigonometric ratios where the one that we are given says cosine theta is equal to negative seven divided by 25 theta is in quadrant three. Now they tell us a lot by telling us that data is in quadrant three. So when it comes to cosine and trigonometric ratios in general, we're going to be using kind of a, a plot, a small graph and a right triangle to help us determine the values. Now, before we get to that, a very important rule in trigonometric rats that we should note is the rule of Soka or S O H dash C A H dash T O A. Now, in this rule, we have three different three letter abbreviations. The first letter of each abbreviation stands for either sine cosine or tangent S C or T. So in this case, we're going to focus on K C A H because we're dealing with cosine. So we'll have that cosine. So the first letter cosine data is equal to the middle letter divided by the third letter. Now, the A will stand for adjacent, the H will stand for hypotenuse and the O will stand for opposite. So if we're dealing with cosine, we have C H, we have the first letter C cosine, theta is equal to the second letter A adjacent divided by hypo, the third letter H. So we have that cosine theta is equal to adjacent, divided by hypos. And they told us that that would be equal to negative seven, divided by 25. So we have that the adjacent sides of theta is a seven and the hypotenuse of this right triangle that we will use is 25. So they told us that we're in quit three. So we can draw a Y axis and an X axis with a lot of space on quadrant three. And then we can make a right triangle in quadrant three. So there could be a line coming out of the origin and into the quadrant. And then we'll draw a line connecting that line to the Y axis and making a right triangle. And then we can label our angle theta and plug in the points. We know. So we know that the adjacent side will be uh seven units and the hypotenuse side will be 25. And we can label the remaining side as B. Now, why do I wanna label it as B? Well, when we have a right triangle and we're trying to find out the different sides we use a rule called the Pythagorean theorem. And that Python theorem states that A squared plus B squared equals C squared where A and B are the two different legs of the triangle and C is the hypotenuse. So using that theorem, we can have that seven squared is one leg plus B squared is the other leg is equal to 25 squared or the hypotenuse. So we'll be left with 49 plus B squared equals 625. And if we subtract 49 from both sides, we'll have that B squared is equal to 576. And finally, we take the square root of both sides to get that B is equal to 24. So now that we have all the sides of our triangle, we can go ahead and solve for the other five trigonometric ratios. And before we do that, we should know something about quadrant three, which is where we are. So quadrant three, let's know which trigonometric ratios will be positive quadrant three will have tangent data and co tangent theta those two will be positive in quad and three, the rest of our ratios will be negative. And we'll see now why. So let's start with sine data when you're thinking of sine think of why values. So anything above the X axis are positive Y values and anything below the X axis are negative Y values, we are in quadrant three which is below the X axis. Therefore, we have negative Y values and we have a negative sign. So we have data if we look at. So we have that sin is equal to opposite, divided by hypo. So we have, in our case, sine is equal to negative opposite divided by hypotenuse. So we have negative 24 divided by 25. And that will be our first sign cosine data they gave to us. So we can go ahead and skip to tangent data. Now, tangent data is part of the to A T O A. So we look at tangent is equal to opposite, divided by adjacent. So we have the opposite is 24 divided by the adjacent, which is seven. And remember tangent data will be positive in quadrant three. So that will be the answer for that. And we can move on to cos data sequent theta is equal to one divided by sine theta. So we just uh get the y, the sign that we calculated and flip it. So we'll have negative 25 divided by 24 as our co co sequent answer. And we can move on to C A theta where sequences can theta is equal to one divided by cosine theta. So we get the, that, that they gave to us and we flip it as well. So we have negative 25 divided by seven. And finally, we have cotangent theta which is equal to one divided by 10 tangent theta. So we just get the tangent data that we calculated and flip it and we'll have that code tangent is equal to seven divided by 24. And once again, that will be positive. So just like that, we've obtained all six trigonometric ratios, we started with the, that they gave us and the quadrant that they provided us for the and we were able to use a bit of graphing and some calculations to calculate the other five trigonometric ratios. 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 = -3/5, θ in quadrant III

Key Concepts

Trigonometric Functions

Quadrants of the Unit Circle

Pythagorean Identity

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