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

Question

Use identities to solve each of the following. Rationalize denominators when applicable. See Examples 5–7. Find cot θ , given that csc θ = ―1.45 and θ is in quadrant III.

Accepted Answer

Hello, everyone. We are asked to solve using identities and rationalize the denominator. If applicable, we want to determine the tangent of theta if the C can of theta equals negative 1.25. And theta is in quadrant three, our answer choices are a negative 0.25 B 0. C negative 0. D 0.75. So first, if we're using identities, thinking about an identity that has both tangent and C can I recall the identity one plus tangent squared of theta would equal the C can squared of theta. So we're gonna use this and we're gonna solve for tangent to find out what the value would be. So plugging in what we know we have one plus the tangent squared of theta equals, we know the CK A theta is negative 1.25. So we're gonna square that. So when I do, I have one plus the tangent squared of theta equals 1. to get tangent squared by itself, I'm subtracting one from both sides. So the tangent squared of theta would equal 0. and then I'm gonna take the square root of both sides to get tangent by itself. And I get the tangent of theta is equal to 0.75. And technically, it's plus or minus. So let's find out if in quadrant three, thinking about our quadrants, we know in the first quadrant all are positive. Second quadrant sine is positive. Third quadrant tangents positive fourth quadrant cosines positive. So if we're in the third quadrant tangent will be positive. So we don't need the plus or minus anymore. We can say that the tangent of theta equals 0.75 and it is a positive 0. and that's answer choice. D have a nice day.

Use identities to solve each of the following. Rationalize denominators when applicable. See Examples 5–7. Find cot θ , given that csc θ = ―1.45 and θ is in quadrant III.

Key Concepts

Reciprocal Trigonometric Identities

Sign of Trigonometric Functions in Quadrants

Pythagorean Identity and Rationalizing Denominators

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