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. 
 
 cos θ = ―5/8 , and θ is in quadrant III

Accepted Answer

Hello, everyone. We are asked to determine the value of the other five trigonometric ratios. We are given that the cosine of theta equals negative 3/7 and that the lies in quadrant three. So first, let's take apart the cosine of data a little bit. So it's a cosine of data is negative 3/7. So I recall that when I'm working with trig ratios cosine is the ratio of X to R. And I recall that I can find my missing value by using the Pythagorean theorem so that X squared plus Y squared would equal R squared. So since I do need to find the value of why I'm gonna say based on the cosine theta that X is negative three, why is our unknown for now? We'll find it in a moment and R equals seven. So I'm gonna use the Pythagorean theorem to solve for Y. So I'll have negative three. I put it in parentheses to group the negative with the three squared plus Y squared equals R which is seven squared, three squared is nine plus Y squared equals 49 minus nine from both sides. I get Y squared equals taking the square root of both sides and get Y equals plus or minus the square of 40. But I want to simplify that. Um And first, because we're in quadrant three, I know Y is gonna be negative. So we'll keep that up in our answer. And I'm gonna simplify the square to 40 which I know I can break down by factors into the square root of four, multiplied by the squared of 10 square to four is two. So this would be two radical 10. So our Y value is gonna be a negative two radical 10. So let's see if we can use that to find the rest of our ratios. So now the sign of data, so cosine is X divided by R sine is Y divided by R. So sign would be negative two radical 10 divided by R which is seven. So there's one of them, the sin of theta is negative two radical 10 divided by seven. Next tangent of data. So tangent is Y divided by X. So why is negative two radical 10 divided by X which was negative three negative divided by a negative is a positive. So we'd have two radical 10 divided by three. And that makes sense because in quadrant three tangents positive and if we're in quadrant three, we should have a positive tangent. So now the next three are a little trickier just because they aren't usually the ones that stand alone, they go off of another one. So I'm gonna start with the sea can of theta. And I recall that the CK of theta is the same as one divided by the cosine of theta, which is also sometimes referred to as the reciprocal of the cosine. So this would be R divided by X. So R is seven X is negative three. So we can say this is negative seven thirds and that is our C can of theta for the can of theta Coy can is one divided by the sign of theta. Again, it's also sometimes looked at as the reciprocal of sign. So this would be R divided by Y R is seven, Y is negative two radical 10. We will need to rationalize this denominator. So we're going to multiply both top and bottom by radical 10. And when we do our numerator is seven radical divided by. So we have negative two and that's gonna be multiplied by the square to 10, multiplied by the square to 10, the square to 10, multiplied by the square to 10 is 10. So 10 multiplied by negative two is negative 20. So we can say this is seven radical 10 or negative seven radical 10 divided by 20. So that is our can of data and we have one more, we need to do the coat tangent. And I'm actually gonna put that over here a little bit. So it's out of line with the others. But I have a little bit more work space. So the cotangent of theta is one divided by the tangent of the which is also sometimes referred to as the reciprocal of the tangent. So this would be X divided by Y so X is negative three. Why is negative two radical 10? I again need to rationalize my denominator by multiplying numerator and denominator by radical 10. And when I do so I have negative three radical divided by negative 20 double negative. So a negative divided by a negative is a positive. This would be three radical 10 divided by as our cotangent of the. So we now have all five of our ratios. Remember rationalizing the denominator and make sure you check your signs. Have a nice day.

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Give all six trigonometric function values for each angle θ . Rationalize denominators when applicable. cos θ = ―5/8 , and θ is in quadrant III