Sketch an angle θ in standard position such that θ has the lea...

Question

Sketch an angle θ in standard position such that θ has the least positive measure, and the given point is on the terminal side of θ. Then find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. See Examples 1, 2, and 4. (―2√3 , 2)

Accepted Answer

Hello, everyone. We are asked to plot the least positive measure of the given angle in standard position such that the given point is on the terminal side. Also, we want to write the value of all trigonometric functions. For the given angle, we will perform rationalization wherever necessary. We're given the point negative three radical two radical seven. So I'm going to begin by plotting that on our coordinate plane, we want to plot the least positive measure from the standard position. So we're gonna start with this point because we are told this point is on the terminal side to plot negative three radical two. I used a calculator and approximated it to negative 4.243 and I approximated radical 7 to 2.646. This just helped me find it a little easier on my coordinate plane. So I'm a little bit past negative four on the negative X axis and a little bit less than three on the positive Y axis. So this point is in quadrant two, I'm labeling the point negative three radical two radical seven connecting it to the origin. I'm going to draw an arrow just to show the rotation and label my angle as the. So next, I want to write the value of all the trigonometric functions. So I recall that to do that, I need to know the value for X, the value for Y and the value for R. Well, right now, looking at the point we're given, I can say X is negative three radical two. And why is radical seven? But I do not know R yet. Luckily we have a formula for that. If you recall R equals the square root of X squared plus Y squared. So this will equal the square root of negative three radical two squared plus radical seven squared. So the square root and then negative three radical two squared, negative three squared is nine radical two squared is two, nine multiplied by two is 18 plus the square root of seven squared is seven. So this will be the square root of 25 which is five. So RR value is five. So now let's find all of those ratios. So we're going to find those values. We're going to start with the sign of theta. So recall that sin is Y divided by R. So our Y value is the square root of seven. Our R value is five. So our sine of theta is radical seven divided by five, no rationalization needed. Next, the cosine of theta is X divided by R. So X is negative three radical two divided by R which was five. And again, no rationalization needed negative three radical two divided by five tangent of data is Y divided by X. So Y is radical seven divided by X which is negative three radical two. I will need to rationalize the denominator here. So I'm multiplying numerator and denominator by the square root of two, the square root of seven multiplied by the square root of two is the square root of 14 and negative three radical two, multiplied by radical two is gonna give me negative six. So I can leave that negative out front in the numerator or in the denominator, whichever works. I'm gonna put it out front, put a negative open a set of parentheses and then put the square to 14/6 in the parentheses. And that's gonna be my tangent of data. Next, I'm gonna go do the Koi can of theta Kosi can is R divided by Y. So five divided by the square root of seven. I will need to rationalize this one. So I'm going to multiply numerator and denominator by the square to seven. So I end up with five radical 7/7. So that is the Kosi can of theta five radical seven divided by seven. The C can't of theta will be R divided by X. So R again is five, X is negative three radical two. We do need to rationalize this denominator. We'll do that by multiplying numerator and denominator by the square root of two. So my numerator is five radical two, my denominator ends up being negative six. And again, I can rewrite this with the negative out front. So I'll put a negative and then open parentheses, five radical two divided by six in the parentheses. And that is my C can of theta finally, the cotangent of data and that is X divided by Y. So X is negative three radical two divided by Y which is radical seven. This needs to be rationalized. So we'll multiply numerator and denominator by the square root of seven. And I end up with negative three radical 14 in my numerator over seven. And again, I can either leave the negative there or put it out front and put the fraction in a set of parentheses. So I either have a negative and then open parentheses. Three radical 14 divided by seven closed parentheses. That is my cotangent of data. And in the end, we have all six trig functions and we drew where the angle is. So we are all set. Have a nice day.

Sketch an angle θ in standard position such that θ has the least positive measure, and the given point is on the terminal side of θ. Then find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. See Examples 1, 2, and 4. (―2√3 , 2)

Key Concepts

Standard Position of an Angle

Finding the Reference Angle and Quadrant

Six Trigonometric Functions from Coordinates

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