Solve each equation for exact solutions.
2 arccos (x/3 -...

Question

Solve each equation for exact solutions.

2 arccos (x/3 - π/3) = 2π

Accepted Answer

Hey, everyone here we are asked to find the exact solution of the equation seven multiplied by R cosine or cosine inverse of X divided by five minus pi divided by five is equal to seven multiplied by pi, we have four answer choice options. A X is equal to pi BX is equal to pi minus five CX is equal to five multiplied by pi and answer D no solution. So to find the exact solution of our given equation, we need to first solve for X. And so to solve for X, we first need to get rid of the coefficient of our cosine or cosine inverse. And so that means we just need to divide both sides of the equation by the coefficient. So we divide both sides by seven. And now rewriting, we have cosine inverse of X divided by five minus pi divided by five is equal to pi. And now we need to get rid of the cosine inverse function. And so that means we just need to take the cosine of both sides. And in doing this, the left hand side, we see that cosine and cosine inverse cancel each other out. And So we are left with X divided by five minus pi divided by five is equal to cosine of pi. Now, our next step in solving four X is to just move over this negative pi divided by five from the left side to the right hand side. And so we just need to add pi divided by five to both sides of the equation. And so we have X divided by five is equal to cosine of pi plus pi divided by five. And now the last step is to get rid of the denominator for X. And so that means we just multiply both sides of the equation by five. And in doing this, we have X is equal to five multiplied by cosine of pi plus pi. And now that we have solved four X, we just want to simplify the right hand side and to do this, we need to recall the unit circle. And so from the unit circle, we find that cosine of pi is equivalent to negative one. So now just substituting in this value into the right hand side, we have X is equal to five, multiplied by negative one plus pi. And now just simplifying and rewriting, we find that X is equal to pi minus five. And with this final solution, we have found the exact answer or the exact solution of our given equation. And so we are left with answer choice B where again, we found that X is equal to pi minus five. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Solve each equation for exact solutions.
2 arccos (x/3 - π/3) = 2π

Key Concepts

Arccosine Function

Solving Trigonometric Equations

Exact Solutions

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