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.

tan θ = -2/3, sin θ > 0

Accepted Answer

Everybody I have read. All right today that we're going to be looking at this question that states determine the value of the five other trigonometric ratios where the trigonometric ratio that they give us is tangent theta is equal to negative nine divided by 40. And they also let us know that sign of theta is greater than zero. Now, we're looking at this question, we have to ask ourselves, well, they, they give us tangent data but what exactly is tangent data? Well, tangent theta is equal to sign of theta divided by cosine of theta. And to continue to solve for the other five trigonometric ratios, we're going to be using a graph. We want to see where our data is and the size of each side of our graph. And to do that, we have to see what part and what quadrants we are in. So if we have 10 data is equal to sine data divided by cosine data, we know that therefore tangent the will be negative and we're focusing on the negative because they tell us that our tangent data is negative. So we know that we will be negative in quadrants two and four. Now we know this because for tangent data to be negative sign of theta and cosine data must be opposite signs. So one of them has to be positive and the other one has to be negative and this occurs in quadrants two and four. Now they tell us that sin of theta is greater than zero. So it's positive. Now sign of fata is positive in two quadrants and that will be in quadrants. Well, when you think about sign, you want to think about the Y axis, right? So Sine is po when is the Y axis positive? Well, the Y axis is positive above the X axis and what two quadrants are above the X axis. Well, tho those are quadrants one and two. So Sine is positive in quadrant one and two. So now we look at tenant data and the sign data that we have and see what quadrant they have in common in this case, that will be quadrant number two. So we know that we will be in dealing with quadrant number two. Now, before we start drawing and, and using our graph, let's remember a very important rule in trigonometric ratios, which is so and that is S O H dash C A H dash T O A. Now this is kind of an acronym that I'm going to explain a bit better. So the first letter of each three letter abbreviation stands for either Sine cosine or tangent in our case, I'm going to be given an example of the to at the end T O A. So T stands for tangent, we have tangent data, the O stands for opposite the A stands for adjacent and the H would stand for hypotenuse if we were using a different abbreviation because we're using to, we are only dealing with the O and the A. So we have to think the first letter in this case T is tangent equals the last two letters divided by each other. So the first one being oh here tangent is equal to O which is opposite, divided by the second letter which is adjacent. And now this is equal to negative nine divided by 40. So now we know that the opposite side of our data angle will be nine and the adjacent side will be 40. And we can use that information to draw a small graph using this information. So we draw a Y axis and an X axis and leave an emphasis on the second quadrant since that's where we're going to be working. And we can draw a hypo coming out of the origin and into quadrant two. And then a straight line from the hypotenuse connected to the X axis to create a right triangle. And then we can put our data angle and we can plug in the points that we know. So we have that opposite side of the data angle will be nine and the adjacent side would be 40. And the hypotenuse side will label as C because what do we use when we have a right triangle? And we want to figure out the measurement of the sides, we use Pyar theorem. And that is A squared plus B squared A and B B in the legs of the triangle equals C squared, where C is the hypos. So we'll have nine squared plus 40 squared equals C squared or 81 plus 1600 is equal to C squared or 1681 is equal to C squared. And if we take a square root of both sides, we'll have that 41 equals C. So now before we start calculating the other trigonometric ratios, now that we have the length of each side, we should note that in quadrant two only sign of data and cos theta are positive. So every other trigonometric ratio will be negative. So let's start with sign of theta sign of theta we look at so we'll have that s sign of theta is equal to O which is opposite, divided by H which is hypo. So the opposite side we said was nine divided by the hypos, which we calculated was 49. We required to this will stay positive. And now we move to cosine theta cosine data will be equal to adjacent, divided by hypo. And since we're in quadrant two will be negative. Because when you're thinking about cosine, think about the X axis. When is the X axis positive and when is it negative? Well, it's negative to the left of the Y axis and the quadrant two is to the left of the Y axis. So our cosine data in this case will be negative. So we'll have negative adjacent over hypotenuse which is divided by 41. And then we can move on to sequence data. You theta is one divided by a sine theta. So we just take the reciprocal of the sine data that we calculated. So we'll have that cos second data is equal to 41 divided by nine. And then we move on to sequence data CCAT theta is one divided by cosine theta. And remember with co sequences, we'll have co sequence will be negative as well as the second that we'll calculate now. But let's remember the negative sign where it goes. So second data is equal to one divided by cosine data and that will be equal to negative the reciprocal of the cosine data that we calculated. So we'll have negative 41 divided by 40. And finally, we'll have called tangent theta which is equal to one divided by tangent theta. And the tenant data that they gave us at the beginning was negative nine divided by 40. So we just take the reciprocal of that and we'll have negative 40 divided by nine. So just like that, we have all six trigonometric ratios where they gave us the tangent of data and using our small graph and our small measurements, we're 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. tan θ = -2/3, sin θ > 0

Key Concepts

Trigonometric Functions

Quadrants of the Unit Circle

Pythagorean Identity

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