In Exercises 99–104, find two values of θ, 0 ≤ θ

Question

In Exercises 99–104, find two values of θ, 0 ≤ θ < 2𝜋, that satisfy each equation._sin θ = - √22

Accepted Answer

Hello, everyone. We are asked to determine two possible solutions for theta where theta is between zero and two pi or it could be zero that fulfill the following equation. The tangent of theta equals negative radical three divided by three. We have four answer choices. A theta equals five pi divided by six and 11 pi divided by six B. Theta equals seven pi divided by six and 11 pi divided by six C. Theta equals five pi divided by six and seven pi divided by six D. Theta equals pi divided by six and 11 pi divided by six. OK. To begin with, we are told the tangent of theta equals negative radical three divided by three. So the tangent is negative and if the tangent is negative, this means we are working in quadrant two and quadrant four only those are the places where tangent is negative. So working backwards a little bit, we have the tangent of theta equals negative radical three divided by three. I can say that the arc tangent of radical three divided by three. I don't need to keep the negative because I know we can only be in quadrants two or four will equal theta. So the arc tangent of radical three divided by three using my unit circle. This will tell me that theta equals pi divided by six. However, because we are working in quadrant two, we need to treat the tangent of theta as if it's the tangent of pi minus the. So to find that out, we will do pi minus the theta value we just found. So pi divided by six. And when we do so we find one of our values and that is five pi divided by six, that'll be for quadrant two and for quadrant four to find the tangent of theta because we have so many co terminal angles, we need to do the tangent of two pi minus the again, this is for quadrant four. So two pi minus the theta we found initially. So pi divided by six, this will give us 11 pi divided by six. So these are our two values or up to possible solutions for tangent of the equaling negative radical three divided by three. And that's answer choice. A theta equals five pi divided by six and pi divided by six have a nice day.

In Exercises 99–104, find two values of θ, 0 ≤ θ < 2𝜋, that satisfy each equation._sin θ = - √22

Key Concepts

Sine Function and Its Range

Unit Circle

Finding Angles in Trigonometric Equations

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