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.

2 tan θ sin θ - tan θ = 0

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 two multiplied by cotangent squared of theta multiplied by cosine of theta minus cotangent squared of theta is equal to zero. Here we have four answer choice options. Answer A, the solution set of 60 degrees, 90 degrees, 270 degrees and 300 degrees. B the solution set of 90 degrees, 270 degrees and 300 degrees. See the solution set of 60 degrees and 90 degrees and answer D the solution set of 60 degrees, 270 degrees and 300 degrees. So to begin this problem, we want to start by simplifying our given equation as much as possible. So looking at the left hand side of the given equation, we notice we have two terms which both share cotangent squared of theta. So this tells us we can start by fact strain out cotangent squared of theta from both terms. And so we now have cotangent squared of theta multiplied by the quantity of two multiplied by cosine of theta minus one is equal to zero. And now we can notice we are multiplying together two expressions which are set equal to zero. And so this tells us, we need to recall the zero factor property which allows us to rewrite our current equation as two individual equations. And so with the zero factor property, we know that cotangent squared of theta can be equal to zero or that two multiplied by cosine of theta minus one can be equal to zero. So now that we have our two simple individual equations, we just need to solve each equation for their respective trigonometric identity. So looking back at cotangent squared of theta is equal to zero, we need to solve for cotangent of theta. So all we need to do here is take the square root of both sides. And so we now find that cotangent of theta is equal to zero. And with this, we can move on to our second equation where again we have two multiplied by cosine of theta minus one is equal to zero. Here solving for cosine of theta, we first just add one to both sides. And so we have two multiplied by cosine of theta is equal to one. Now, we just need to divide both sides by the coefficient of cosine which is two. And so we find that cosine of theta is equal to one half. And now utilizing our unit circle, we need to find the respective theta values which allow both of our current statements to be true. So looking back at our cotangent of theta is equal to zero, to more easily identify our theta value on the unit circle. We first just want to recall that cotangent of theta is equivalent to cosine of theta divided by sine of theta. And with this expression, we can find on our unit circle that our theta value which allows cotangent of theta to be equal to zero are for when theta is equal to either 90 degrees or 270 degrees. So now we have found two solutions from our one equation. So moving on to our second equation where again, we have cosine of the is equal to one half. Well, here using the unit circle, we know that these, the values which allow cosine of data to be equal to one half are when the is equal to 60 degrees or 300 degrees. So now we have found a total of four different possible solutions for theta for our given equation. So we just want to summarize our findings by writing our solutions as part of a solution set. And so here we have the solution set of 60 degrees, 90 degrees, 270 degrees and 300 degrees. And with this final solution set, we are left with answer choice A where again, we have the solution set of 60 degrees, 90 degrees 270 degrees and 300 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.
2 tan θ sin θ - tan θ = 0

Key Concepts

Trigonometric Equations

Factoring and Zero-Product Property

Solving for Angles in a Given Interval

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