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Given tan x = -5⁄4, where π/2

Question

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Given tan x = -5⁄4, where π/2< x < π, use the trigonometric identities to find cot x, csc x and sec x.

Accepted Answer

Hello, everyone. We are asked to find the exact values of the cotangent of theta. Kana. Theta and C can theta, if the tangent of theta equals negative 25 divided by 24 and theta terminates in the second quadrant, we will use trigonometric identities. Answer choice A K tangent of theta equals 24 25th. Kosic can of theta equals the square root of 1201 divided by 24. The C Canna theta equals negative square root of 1201 divided by 25 B. The cotangent of theta equals negative 24 25th. The Kan of Theta equals the square root of 1201 divided by 25. The C of theta equals negative square root of 1201 divided by 24 C. The cotangent of theta equals 7 25th. The Kosi can of theta equals the square root of 771 divided by 24. The C can of theta equals negative square root of 771 divided by 25 D. The cotangent of theta equals negative 7 25th. The Kant of theta equals the square root of 771 divided by 24. The C can of theta equals negative square root of 771 divided by 25. OK. So I'm gonna start with the one that seems quickest to solve. And that would be the cotangent of data. So we know that the tangent of theta is negative 2524. And through our reciprocal identity, we know that cotangent of theta will equal one divided by the tangent of theta. And when we do, so we get the cotangent of theta equals negative 24 25th. So we found one of our answers already that the cotangent of theta is negative 24 25th. Now let's use that to find the other two. So I will go to the um let's go to the C can next. So we have the C can't squared of theta equals the tangent squared of theta plus one. So that's a Pythagorean identity. And I think that will help us here. So we don't know the C can't square to theta. So leave that alone for now, we do know tangent is negative 25 24th. So I'm gonna square that, then we'll add one. So the C can't squared of theta is going to equal a positive 625 divided by 576 plus one. So then we have the C squared of theta equals and I'm gonna treat one as 576 divided by 576. So I combine those and I get 1201 divided by 576. Since we are working with the C can squared, but we only want the C can't, I'm gonna take the square root of both sides. And since we are in quadrant two, I know that this will ultimately end up being a negative. So I'm gonna put the negative there. So the C can of theta equals negative. And I'm gonna take the square root of both the numerator and the denominator here. And that will give me negative square root of 1000 201 divided by 24. So our C of theta is negative square root of 1201 divided by 24. All right. So one more to find the Koi Camp, I'm gonna use another Pythagorean identity. So we're gonna say the KC can't squared of theta equals one plus the cotangent squared of theta. All right. We're gonna use the cotangent that we found. So we're saying the co C can't squared of theta equals one plus. And in parentheses, I'm gonna put negative 24 25th squared. So this will give us the cosecant of theta squared equals one plus and when we square negative, it becomes positive. So this will be 576 divided by 625. And now I want to combine those. So I'll have the KC can't squared of theta equals 1201 divided by 625 and taking the square root of both sides because I don't want the Kosi can't squared, I just want the Kosi can't. So the kant of theta equals and if I take the square root of both the numerator and the denominator, I will get the square root of 1201 divided by 25. And since we're in quadrant two, this is positive. So we can say the kant of theta is equal to the square root of 1201 divided by 25. So now checking my answer choices, the answer choice that matches all three of these values is answer choice B have a nice day.

Work each problem.
Given tan x = -5⁄4, where π/2< x < π, use the trigonometric identities to find cot x, csc x and sec x.

Key Concepts

Trigonometric Identities

Quadrants and Signs of Trigonometric Functions

Reciprocal Trigonometric Functions

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