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University of Southern California

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Problem 34

Solving a Trigonometric Equation In Exercises $31-34$ , find two solutions of each equation. Give your answers in radians $(0 \leq \theta \leq 2 \pi) .$ Do not use a calculator.

$$\begin{array}{l}{\text { (a) } \sin \theta=\frac{\sqrt{3}}{2}} \\ {\text { (b) } \sin \theta=-\frac{\sqrt{3}}{2}}\end{array}$$

Answer

(a) $\theta=\frac{\pi}{3}$ or $\frac{2 \pi}{3}$

(b) $\theta=\frac{4 \pi}{3}$ or $\frac{5 \pi}{3}$

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## Discussion

## Video Transcript

problem. Number 34 asks us to find two solutions for each of the following equations. And then to give our answers in radiance in between zero and two pi and finally were asked to not use a calculator. So, essentially, we're gonna have to leverage our knowledge of the unit circle toe, identify values of theta where sign of that fatal equals square root three over two. Now, just based on our knowledge of the unit circle, we know that at pie thirds and two high thirds, we have 1/2 for our co sign in square root. Three over two for our sign. And then over here, two brothers. You have negative 1/2 for co sign and square root three over two for our sign. Right? That's because this X distance right here that corresponds this value in this. Why? Distance up here corresponds to the sign value of our thing. Okay. And so that's the answer for party. Um, data equals hi thirds and two pi thirds. No, her part b. We see that we're looking for a place on our unit circle were a sign of that. Fatal equals negative square root be over too and, you know, up here on this circle, this wide distances square be over too. So negative square, if there were two, would be the same distance down. And that's exactly where value will occur. All right. And so if this is a full pie adding an extra third, it's gonna be four pi thirds. And this over here, it's gonna be five pi thirds at four pi. Thirds are cosign. Value is negative 1/2 and our sign value is negative. Square root three over two and five pi thirds at exact co sign. Value is 1/2 and the sign value is negative. Square of three over two. No, that's our answer. Fada equals for pie thirds and five by thirds. Now, some of you might have been wondering, you know about other values honor unit circle And how we know that those aren't you? No, corresponding to this sine theta. Well, let's just take a quick look. It, for example, High sixth. Well, hi six. This the coastline value is gonna be square. It be over too. And our signed values gonna be 1/2 at pie sticks. You see that behavior happen over here? You know what? Five pie six we're co sign equals negative square three over to sign equals 1/2 and on and on and off Throughout the unit circle, you'll find that the only places or sine theta equals square through over to visit pie thirds and to buy thirds. And the only places were sign of data equals negative. Three negative square root. Three over two. Is it four point thirds and fight by thirds. So there you have it, trying to circle this right here. And there you go.

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