Solve each equation for exact solutions.
cos⁻¹ x = sin⁻¹...

Question

Solve each equation for exact solutions.

cos⁻¹ x = sin⁻¹ 3/5

Accepted Answer

Hey, everyone. Here we are asked to find the exact solution of the equation. Cosine inverse of X is equal to sine inverse of 12 divided by 37. Here we have four answer choice options. A X is equal to 35 divided by 37 BX is equal to 37 divided by 35 CX is equal to 12 divided by 35 and answer D no solution. So to begin this problem, we want to start by solving our given equation for X and looking at our equation, we notice that the right hand side is rather complex. So we can begin by simplifying this equation by letting the variable U be equivalent to the right hand sign. So U is going to be equal to sine inverse of 12 divided by 37. And so now we just want to rewrite our given equation with this identified variable U. And so we have cosine inverse of X is equal to U. And now we can much more easily solve for X. All we need to do is take cosine of both sides. And now we have cosine and cosine inverse, we'll cancel each other out leaving us with just X is equal to cosine of U. And so now we want to start solving our equation since we have already solved for X. And we do this by referring back to our identified variable U, recalling that U is equal to sine inverse of 12 divided by 37. Well, this tells us that if we have S of U, this will be equivalent to 12 divided by 37. And with this information, we can work towards finding cosine of U by drawing first our right triangle. So our right triangle, we have our 90 degree angle as well as our angle U. And now to start filling in the borders of this triangle, you need to recall that sign is equivalent to opposite the opposite side divided by the hypotenuse. And so we know that the opposite side of angle U will be 12. And so the hypotenuse is then 37 and we will let the adjacent side be variable A. So now we want to solve four variable A. So we need to recall the Pythagorean theorem which states a squared plus 12 squared is equivalent to 37 squared. And now just solving for A, we have A is equal to the square root of 37 squared plus 12 squared. And we can notice here that we will only have a positive square root since length, the length of the adjacent side can only be positive. So now just evaluating this expression for A, we find that A is equal to 35. And so now that we know that the adjacent side is 35 we can find cosine of U. So first, we need to recall that cosine is equivalent to adjacent, the adjacent side divided by the hypotenuse. So now looking at our right triangle, we know that this is equivalent to a divided by 37 and we found that A is 35. So we know that cosine of U is equivalent to 35 divided by 37. And because this is the equivalent value for cosine of U, we know that this is also the value for X. So we can conclude here that our value for X is 35 divided by 37. And with this, we have found the exact solution for our given equation. And so we are left with answer choice A where again, X is equal to 35 divided by 37. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Solve each equation for exact solutions.
cos⁻¹ x = sin⁻¹ 3/5

Key Concepts

Inverse Trigonometric Functions

Relationship Between Sine and Cosine

Solving Trigonometric Equations for Exact Values

Watch next