Give all six trigonometric function values for each angle θ. Rat...

Question

Give all six trigonometric function values for each angle θ.  Rationalize denominators when applicable. See Examples 5–7. 
 
 csc θ = ―3 , and cos θ > 0

Accepted Answer

Welcome back. I am so glad you're here. We are asked to enumerate the value of the other five trigonometric functions of M given the following trigonometric function and condition rationalize the denominators if necessary. We're given that the kant of M is equal to negative eight with the condition that the cosine of M is greater than zero. So first of all, let's figure out what quadrant we're talking about here. We recall from previous lessons, the acronym, all students take calculus which tells us that in the first quadrant, all trigonometric functions are positive. In the second quadrant, the sign and the Kosi can't are positive in the third quadrant, the tangent and the co tangent are positive. And in the fourth quadrant, the cosine and she cant are positive. So we've got a negative kant. Our cosecant of M is equal to a negative eight. So cosecant is negative in the 3rd and 4th quadrants, we've got a positive cosine, cosine is greater than zero and cosine is positive in the 1st and 4th quadrants. So where are both conditions met? That's going to be in quadrant four. So we're dealing with quadrant four here. All right. Now, what else do we know? We know that when we have a cosecant of an angle equal to negative eight, we can choose any point on the terminal side of angle M. And it's easy to choose an R value of eight. And what we can do with that is knowing that we're somewhere in quadrant four, we are going to have RKC can of M equals negative eight kant is found by taking R divided by Y, we have R divided by Y equals negative eight. Our R value said we said is going to be eight. So our Y value in order to get this to equal negative eight is going to be negative one. So our Y value is negative one. Now, for our X value, we recall from previous lessons that we've got R squared equals X squared plus Y squared to help us solve for X R is eight. So we've got eight squared equals X squared is unknown plus Y is negative one squared. Eight squared is 64. So 64 equals X squared plus negative one squared is positive one. We'll subtract one from each side and we have 63 equals X squared. Taking the square rut of both sides. We'll have our X equals. Now we have plus or minus the square of 63. Well, 63 is nine multiplied by seven and we can take the square root of nine. So that comes out and that's plus or minus three square root seven is our X. However, we know we're in quadrant four and in quadrant four, our X values are positive. So we can disregard that negative value and our X is only equal to positive three square root seven. All right, now that we have our XY and R, we can pretty quickly figure out those other five trigonometric functions of M. So I'm going to move this work off to the side. And now we'll figure out the other five trigonometric functions starting with S we know that the sign of an angle is going to be Y divided by RY is negative one divided by R is eight. So the sign of M equals negative 1/8 that one's simplified. All right. Now, the cosine of angle M, we know that cosine is X divided by R X is three square root seven divided by R is eight. That one's simplified. So the cosine of M equals three square root seven divided by eight. All right. Next, we'll do the tangent. The tangent of an angle is found by taking Y divided by XY is negative one divided by X is three square root seven. This one we do need to rationalize. So we'll multiply both the top. Let me rewrite that we will multiply both the numerator and the denominator by the square rep of seven. So in the numerator, we'll have a negative square root seven divided by. And now we have three multiplied by square root of seven, multiplied by square root of seven is just seven. So three multiplied by seven is 21. So there's our tangent M equals negative square root seven divided by 21. All right. For the tangent, excuse me, for the cotangent of M, we know the cotangent is found by taking X divided by Y X. Here is three square root seven, Y is negative one that simplifies to negative three square root seven. So our cotangent of M is negative three square root seven. And then lastly, we have our C can of M C is found by taking R divided by X R. Here is eight divided by X is three square root seven. We do need to rationalize that. So we'll multiply both the numerator and the denominator by square root seven. And in the numerator, we'll have eight square root seven in the denominator just like last time, three square root seven multiplied by seven is 21. So RC can of M is equal to eight square seven divided by 21. And there are our other five trigonometric functions of M. Well done. We'll catch you on the next one.

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Give all six trigonometric function values for each angle θ. Rationalize denominators when applicable. See Examples 5–7. csc θ = ―3 , and cos θ > 0