Solve each equation for exact solutions.
tan⁻¹ x - tan⁻¹...

Question

Solve each equation for exact solutions.

tan⁻¹ x - tan⁻¹ (1/x ) = π/6

Accepted Answer

Welcome back. I am so glad you're here. We are asked to find the exact solution of the equation. Our given equation is the inverse sine of X plus the inverse Kent of one divided by X equals pi divided by three. Our answer choices are answer choice, a zero answer, choice B one half, answer choice C negative one half and answer choice D zero and one half. All right. So we've got an inverse sign being added to an inverse cos can and we recall from previous lessons that sign and cos can are reciprocal functions of each other. So what happens when they're inverses? Well, we can do a little bit of work with that. If we've got the inverse sine of X, we can set that to equal to an angle theta. And we recall from previous lessons on inverse functions that that would be the same as saying that the sine of theta is equal to X. And we recall from previous lessons that the sign of theta is equal to one divided by its reciprocal function cos theta. But we already know that the sine of theta is equal to X. So we can set one divided by the KC can of theta equal to X. If we solve for the KC can of theta, it would equal one divided by X. And if we get the KC can back to its inverse function, that would be saying that the inverse kant of that right hand side one divided by X is equal to theta. And here because the inverse sign of X and the inverse kant of one divided by X are both equal to theta, they're both equal to each other. So now we can use substitution where we've got the inverse kant of one divided by X, we can plug in what it's equal to which is the inverse sine of X. So bringing this down, we have the inverse sine of X plus. The inverse sine of X equals pi divided by three inverse sine of X is a like term with itself. So now we have two inverse sine of X equals pi divided by three. We can just divide both sides by two and we'll have the inverse sine of X equals pi divided by three, divided by two is pi divided by six. So the inverse sine of X equals pi divided by six. And we recall that that would be the same as saying that the sign of pi divided by six is equal to X. Now, we just need to know what is the sign of pi divided by six equal to. We can look back at previous lessons. And the unit circle, we can look at the charts in the textbook, the sine of pi divided by six is the same as pi uh the s of 30 degrees and that's equal to one half. So our X value here is equal to one half. We look at our answer choices and this matches with answer choice. B well done. We'll catch you on the next one.

Solve each equation for exact solutions.
tan⁻¹ x - tan⁻¹ (1/x ) = π/6

Key Concepts

Inverse Tangent Function (arctan)

Tangent Difference Identity for Inverse Tangents

Solving Trigonometric Equations for Exact Values

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