Solve each equation over the interval [0°, 360°). Write soluti...

Question

Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.

sec² θ tan θ = 2 tan θ

Accepted Answer

Hey, everyone here we are asked to find the solutions of the given equation over the interval from open bracket zero degrees to 360 degrees closed parentheses. Here we are given the equation cos squared of theta multiplied by cotangent of theta is equal to four multiplied by cotangent of theta. Here we have four answer choice options. Answer choice A has the solution set containing four different degree values. Answer B contains six different degree values and answers C and D have five different degree values. So to begin this problem, we want to start by simplifying our given equation as much as possible. And so we can begin by moving the term on the right hand side to the left hand side of the equation. And so that means we just need to subtract or multiplied by cotangent of theta from both sides of the equation. And so we now have cos squared of theta multiplied by cotangent of theta minus four multiplied by cotangent of theta is equal to zero. And now we can notice both of our terms on the left hand side, share cotangent of theta. So this tells us that we can factor out cotangent of theta from both of these terms. And so we have cotangent of theta multiplied by the quantity of Kent squared of theta minus four is equal to zero. And now, with our current equation, we notice that we are multiplying two expressions and the right hand side is just zero. So this tells us we can recall the zero factor property which allows us to rewrite our current equation into two individual equations where we now know that cotangent of theta can be equal to zero or coin squared of theta minus four can be equal to zero. So now we just want to solve both of our equations for their respective trigonometric function. And we can notice our first equation is already fully simplified and we just have cotangent of theta is equal to zero. So moving back to our second equation again, we have cosecant squared of theta minus four is equal to zero here to start solving four cosecant of theta. We first just want to add four to both sides of the equation. And so we have cos squared of theta is equal to four and now continuing to solve or coking of beta. Our next step is to just take the square root of both sides where since we are adding the square root, on the right hand side, we need to include a plus or minus in front of the root. And so on the right hand side, we have plus or minus the square root of four. And now simplifying, we end up with two additional equations here where we know that cosecant of theta can be equal to positive two or cosecant of theta can be equal to negative two. And now with all of our equations fully simplified, we can recall the unit circle to find our theta values. However, to make our theta values more easier to identify, we can recall that cotangent of theta is equivalent to cosine of theta divided by sine of theta. And then we also can recall that coin of theta is equivalent to one divided by sine of theta. And so now identifying our theta values on the unit circle, we know that cotangent of theta is equivalent to zero for when theta is equal to 90 degrees or 270 degrees. And now moving on to our other equations, we know coin of theta is equivalent to positive two for when theta is equal to 30 degrees or 150 degrees. Now cos seeking of theta is equivalent to negative two or when theta is equal to 210 degrees or 330 degrees. And with this, we have found six different degree values that can all be included in our solution. And so we can summarize our findings as including these values in a solution set. And so we have the solution set of 30 degrees, 90 degrees, 150 degrees, 210 degrees, 270 degrees and 330 degrees. And with these final solutions, we are left here with answer choice B where again, we have the solution set of six different degree values which are 30 degrees, 90 degrees, 150 degrees, 210 degrees, 270 degrees and 330 degrees. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.
sec² θ tan θ = 2 tan θ

Key Concepts

Trigonometric Identities

Solving Trigonometric Equations

Interval and Solution Set for Angles

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