Solve each equation for exact solutions.
arcsin x = arct...

Question

Solve each equation for exact solutions.

arcsin x = arctan 3/4

Accepted Answer

Hey, everyone here we are asked to find the exact solution of the equation. Arc sine of X is equal to arc tangent of 13 divided by 84. We have four answer choice options. A X is equal to 84 divided by 85 BX is equal to one CX is equal to 13 divided by 85 and answer D no solution. So to begin this problem, we can start by simply rewriting our given equation. So we instead have sine inverse of X is equal to tangent inverse of 13 divided by 84. And so now we want to manipulate this equation to solve for X. And so we can start by simplifying the right hand side since we can notice that we have a rather complex expression. So to simplify, we simply need to let the variable U be equivalent to the right hand side. So U is equal to tangent inverse of 13 divided by 84. So now we just want to rewrite our equation using this identified variable U. So we now have sine inverse of X is equal to U. Now solving for X, we just need to take sign of both sides of the equation. And so now we notice that on the left hand side sign and sign inverse cancel each other out. So we are just left with X is equal to sign of you. And so now we need to find the exact value for a sign of you. And to do this, we need to refer back to our identified variable U where we know U is equal to tangent inverse of 13 divided by 84. And so this expression tells us that if we instead have tangent of you, we know that tangent of you is equivalent to 13 divided by 84. And so now we can utilize this information to work towards finding sign of you. So we first need to draw our right triangle and recalling a right triangle has this 90 degree angle. And we also need to include the angle U. And so now just referring back to our tangent of you, we want to recall that tangent is equivalent to the opposite side of the angle divided by the adjacent side. So this tells us that opposite from angle U, we know that this border, this length here is going to be 13 and then the adjacent side to angle U is 84. And so now we are just left with the hypotenuse, which we will label as variable C for now. And so in order to find sign of U, we want to find the value for C. And so that means we need to recall the Pythagorean theorem which tells us that C squared is equivalent to 84 squared plus 13 squared. And now just solving for C, we have C is equal to the square root of 84 squared plus 13 squared. And now we can notice here, we only have a positive square root since we are dealing with a length which will only have a positive value. So now just evaluating this expression for C, we find that C is equal to 85. And so now that we have found that the hypotenuse C is 85 we can find our value for a sign of U. So we just need to recall that sign is equivalent to the opposite side from the angle divided by the hypotenuse. And with this, we can simply fill in our frown values. And so we find that sign of you is equivalent to 13 divided by 85. And since we have found sign of you, we have also found the value for X. So we know X is equal to 13 divided by 85. And with this, we have found the exact solution for our given equation. And so we are left with answer choice C where again, we found X is equal to 13 divided by 85. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Solve each equation for exact solutions.
arcsin x = arctan 3/4

Key Concepts

Inverse Trigonometric Functions

Relationship Between Sine and Tangent

Exact Values and Right Triangle Ratios

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