Use identities to solve each of the following. Rationalize denomi...

Question

Use identities to solve each of the following. Rationalize denominators when applicable. See Examples 5–7. 
 
 Find csc θ , given that cot θ = ―1/2 and θ is in quadrant IV.

Accepted Answer

Hey, everyone here we are asked to determine seeking of theta if tangent of theta is equal to negative one third. And theta is in quadrant two. And we need to solve using identities and rationalize the denominator. If applicable here, we have four answer choice options, answer choice, a negative square root of 10 divided by three answer B the square root of 10 divided by three answer C negative square root of 10 divided by nine and answer D the square root of 10 divided by nine. So to begin this problem, we first need to recall the Python identity which states that one plus tangent squared of theta is equal to second squared of theta. So now utilizing this identity, we want to substitute in the given value for tangent where again, we are given that tangent of theta is equal to negative one third. So now substituting in this value, we have one plus the quantity of negative one third squared is equal to second squared of theta. And now we just want to start simplifying. So we have one plus one N is equal to second square of theta. And now continuing to simplify, we just need to sum our two values. And so we have that second squared of theta is equal to 10 divided by nine. Now, just solving for CN of theta, we need to get rid of the square by first taking the square root of both sides of our equation. And so we have second of theta is equal to plus or minus the square root of 10 divided by nine. And now we can simplify our fraction underneath the root by recalling that the square root of nine simplifies to three. So simplifying we have second of theta is equal to plus or minus the square root of 10 divided by three. And now we can notice we have this plus or minus in front of the value for sin of theta. So now we need to utilize the other piece of given information where we are told that theta is in quadrant two. And because the is in quadrant two, we know that sin must be negative. And so we can conclude here that sin of theta is equal to negative square root of divided by three. And with this finalized answer, we are left here with answer choice. A where again, second of theta is the negative square root of 10 divided by three. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Use identities to solve each of the following. Rationalize denominators when applicable. See Examples 5–7. Find csc θ , given that cot θ = ―1/2 and θ is in quadrant IV.

Key Concepts

Trigonometric Identities

Cotangent and Cosecant Functions

Quadrants and Angle Signs

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