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# Find the Exact Value

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Find the exact value.

May 10, 2019

#1
+8852
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(a)

$$\sin^2\theta+\cos^2\theta\,=\,1$$       by the Pythagorean Identity.

$$(\frac13)^2+\cos^2\theta\,=\,1$$        because we are given that  $$\sin\theta\,=\,\frac13$$

$$\frac19+\cos^2\theta\,=\,1\\~\\ \cos^2\theta\,=\,1-\frac19\\~\\ \cos^2\theta\,=\,\frac89\\~\\ \cos\theta\,=\,\pm\sqrt{\frac89}\\~\\ \cos\theta\,=\,\pm\frac{2\sqrt2}{3}$$

Since  θ  is in Quadrant II,  cos θ  must be negative. So   $$\cos\theta\,=\,-\frac{2\sqrt2}{3}$$

(b)

$$\sin(\theta+\frac{\pi}{6})\,=\,\sin\theta\,\cos\frac{\pi}{6}+\cos\theta\,\sin\frac{\pi}{6}$$     by the angle sum formula for sin

$$\sin(\theta+\frac{\pi}{6})\,=\,(\frac13)(\frac{\sqrt3}{2})+(-\frac{2\sqrt2}{3})(\frac12)\\~\\ \sin(\theta+\frac{\pi}{6})\,=\,\frac{\sqrt3}{6}-\frac{2\sqrt2}{6}\\~\\ \sin(\theta+\frac{\pi}{6})\,=\,\frac{\sqrt3-2\sqrt2}{6}$$

(c)

$$\cos(\theta-\frac{\pi}{3})\,=\,\cos\theta\,\cos\frac{\pi}{3}+\sin\theta\,\sin\frac{\pi}{3}$$     by the angle difference formula for cos

Now plug in the values for  cos θ,  cos(pi/3) ,  sin θ , and  sin(pi/3)  and simplify. Can you finish this one?

(d)

To use the angle sum formula for tan, let's first find  tan θ.

$$\tan\theta\,=\,\frac{\sin\theta}{\cos\theta}\,=\,\frac{(\frac13)}{(-\frac{2\sqrt2}{3})}\,=\,-\frac{1}{2\sqrt2}\,=\,-\frac{\sqrt2}{4}$$

$$\tan(\theta+\frac{\pi}{4})\,=\,\frac{\tan\theta+\tan\frac{\pi}{4}}{1-\tan\theta\,\tan\frac{\pi}{4}}$$     by the angle sum formula for tan

$$\tan(\theta+\frac{\pi}{4})\,=\,\frac{(-\frac{\sqrt2}{4})+(1)}{1-(-\frac{\sqrt2}{4})(1)}\\~\\ \tan(\theta+\frac{\pi}{4})\,=\,\frac{4-\sqrt2}{4+\sqrt2}$$

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May 10, 2019
#2
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Thanks, man. On d, do you think I need to rationalize the denominator? I know how, but do I need to?

#3
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I don't know either, but I guess it wouldn't hurt. Maybe better to be safe than sorry!

hectictar  May 10, 2019
#4
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Nice, thanks!

#5
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I got the same answer for c as the answer for b. Is that right?