Solve each equation for exact solutions.
4/3 cos⁻¹ x/4 =...

Question

Solve each equation for exact solutions.

4/3 cos⁻¹ x/4 = π

Accepted Answer

Hey, everyone here we are asked to find the exact solution of the equation. Where here we are given three divided by two multiplied by cosine inverse of X divided by two is equal to pi. Here we have four answer choice options. A X is equal to negative one BX is equal to negative square root of two CX is equal to negative one half. And DX is equal to negative square root of three divided by two. In order to find the exact solution of our given equation, we first need to solve our current equation for X. And so the first thing we want to do is get rid of this coefficient for the inverse cosine function. And so that means we need to divide both sides of the equation by the coefficient which is three divided by two. So again, we divide both sides by three, divided by two. Or in other words, we multiply by two divided by three. So doing this, we have cosine inverse of X divided by two is equal to two multiplied by pi divided by three. And now we can notice that our X term is within this inverse cosine function. And so that tells us our next step is to just take cosine of both sides of the equation. So taking cosine of both sides, we notice that cosine and cosine inverse will cancel each other out. And so, so far, we are left with X divided by two is equal to cosine of two pi or two multiplied by pi divided by three. And so now looking back at the left hand side of the current equation, we have two in the denominator. And so our next step is to multiply both sides of the equation by two. And so doing this, we now have X is equal to two multiplied by cosine of two multiplied by pi divided by three. And now that we have solved our equation for X, we can go ahead and evaluate the right hand side. And so the first thing we need to do here is recall our unit circle and from the unit circle, we can find cosine of two multiplied by pi divided by three. And we find that this value or this expression is equivalent to negative one divided by two. And so now we can just substitute in this value into the right hand side of our current equation. And so now rewriting, we have X is equal to two multiplied by negative one divided by two. And so now just multiplying our terms together, we notice that our twos cancel with each other. And so we are just left with X is equal to negative one. And with this final answer, here we are left with answer choice A where again, we found that X is equal to negative one. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Solve each equation for exact solutions.
4/3 cos⁻¹ x/4 = π

Key Concepts

Inverse Trigonometric Functions

Solving Trigonometric Equations

Understanding Radian Measure

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