Exercise (11.3) No (2) Solution


Find the value of θ, 0° ≤ θ ≤ 360° for the following equations. Do not use table.

(a) $ \displaystyle \ \ \ \ \ \ \sin \theta =-\frac{1}{2}$

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$ \displaystyle \therefore \ \ \ \ \text{basic acute angle}=30{}^\circ $

$ \displaystyle \begin{array}{l}\ \ \ \ \ \ \text{But }\sin \theta \ \text{is negative, }\theta \ \text{may be }\\\ \ \ \ \ \ \text{either in the }{{\text{3}}^{{\text{rd}}}}\text{ or }{{\text{4}}^{{\text{th}}}}\text{ quadrant}\text{.}\end{array}$

 $ \displaystyle \therefore \,\ \ \ \theta =180{}^\circ +30{}^\circ \ (\text{or)}\ \theta =360{}^\circ -30{}^\circ \ $

$ \displaystyle \therefore \,\ \ \ \theta =210{}^\circ \ (\text{or)}\ \theta =330{}^\circ $


(b) $ \displaystyle \ \ \ \ \ \ \cos \theta =-\frac{{\sqrt{3}}}{2}$

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$ \displaystyle \therefore \ \ \ \ \text{basic acute angle}=30{}^\circ $

$ \displaystyle \begin{array}{l}\ \ \ \ \ \ \text{But }\cos \theta \ \text{is negative, }\theta \ \text{may be }\\\ \ \ \ \ \ \text{either in the }{{\text{2}}^{{\text{nd}}}}\text{ or }{{\text{3}}^{{\text{rd}}}}\text{ quadrant}\text{.}\end{array}$

$ \displaystyle \therefore \,\ \ \ \theta =180{}^\circ -30{}^\circ \ (\text{or)}\ \theta =180{}^\circ +30{}^\circ \ $

$ \displaystyle \therefore \,\ \ \ \theta =150{}^\circ \ (\text{or)}\ \theta =210{}^\circ $


(c) $ \displaystyle \ \ \ \ \ \ \cos \theta =-\frac{1}{{\sqrt{2}}}$

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$ \displaystyle \therefore \ \ \ \ \text{basic acute angle}=45{}^\circ $

$ \displaystyle \begin{array}{l}\ \ \ \ \ \ \text{But }\cos \theta \ \text{is negative, }\theta \ \text{may be }\\\ \ \ \ \ \ \text{either in the }{{\text{2}}^{{\text{nd}}}}\text{ or }{{\text{3}}^{{\text{rd}}}}\text{ quadrant}\text{.}\end{array}$

$ \displaystyle \therefore \,\ \ \ \theta =180{}^\circ -45{}^\circ \ (\text{or)}\ \theta =180{}^\circ +45{}^\circ \ $

$ \displaystyle \therefore \,\ \ \ \theta =135{}^\circ \ (\text{or)}\ \theta =225{}^\circ $


(d) $ \displaystyle \ \ \ \ \ \ \tan \theta =\sqrt{3}$

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$ \displaystyle \therefore \ \ \ \ \text{basic acute angle}=60{}^\circ $

$ \displaystyle \begin{array}{l}\ \ \ \ \ \ \text{But }\tan \theta \ \text{is positive, }\theta \ \text{may be }\\\ \ \ \ \ \ \text{either in the }{{\text{1}}^{{\text{st}}}}\text{ or }{{\text{3}}^{{\text{rd}}}}\text{ quadrant}\text{.}\end{array}$

$ \displaystyle \therefore \,\ \ \ \theta =60{}^\circ \ (\text{or)}\ \theta =180{}^\circ +60{}^\circ \ $

$ \displaystyle \therefore \,\ \ \ \theta =60{}^\circ \ (\text{or)}\ \theta =240{}^\circ $

(e) $ \displaystyle \ \ \ \ \ \ \tan 2\theta =1$

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$ \displaystyle \therefore \ \ \ \ \text{basic acute angle}=45{}^\circ $

$ \displaystyle \begin{array}{l}\ \ \ \ \ \ \text{But }\tan 2\theta \ \text{is positive, }2\theta \ \text{may be }\\\ \ \ \ \ \ \text{either in the }{{\text{1}}^{{\text{st}}}}\text{ or }{{\text{3}}^{{\text{rd}}}}\text{ quadrant}\text{.}\end{array}$

$ \displaystyle \underline{{\text{For }{{\text{1}}^{{\text{st}}}}\ \text{quadrant}}}\text{,}$

$ \displaystyle \therefore \,\ \ \ 2\theta =45{}^\circ \ (\text{or)}\ 2\theta =360{}^\circ +45{}^\circ \ $

$ \displaystyle \ \ \ \ \ 2\theta =45{}^\circ \ (\text{or)}\ 2\theta =405{}^\circ \ $

$ \displaystyle \ \ \ \ \ \theta =22{}^\circ 3{0}'(\text{or)}\ \theta =\ 202{}^\circ 3{0}'$

$ \displaystyle \underline{{\text{For }{{\text{3}}^{{\text{rd}}}}\ \text{quadrant}}}\text{,}$

$ \displaystyle \ \,\ \ \ 2\theta =180{}^\circ +45{}^\circ \ (\text{or)}\ 2\theta =360{}^\circ +180{}^\circ +45{}^\circ $

$ \displaystyle \ \ \ \ \ 2\theta =225{}^\circ \ (\text{or)}\ 2\theta =585{}^\circ $

$ \displaystyle \ \ \ \ \ \theta =112{}^\circ 3{0}'(\text{or)}\ \theta =292{}^\circ 3{0}'$


(f) $ \displaystyle \ \ \ \ \ \ \tan 3\theta =-1$

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$ \displaystyle \therefore \ \ \ \ \text{basic acute angle}=45{}^\circ $

$ \displaystyle \begin{array}{l}\ \ \ \ \ \ \text{But }\tan 3\theta \ \text{is negative, 3}\theta \ \text{may be }\\\ \ \ \ \ \ \text{either in the }{{\text{2}}^{{\text{nd}}}}\text{ or }{{\text{4}}^{{\text{th}}}}\text{ quadrant}\text{.}\end{array}$

$ \displaystyle \underline{{\text{For }{{\text{2}}^{{\text{nd}}}}\ \text{quadrant}}}\text{,}$

$ \displaystyle \ \,\ \ \ 3\theta =180{}^\circ -45{}^\circ \ $

$ \displaystyle \ \,\ \ \ 3\theta =135{}^\circ \ $

$ \displaystyle \ \,\ \ \ \theta =45{}^\circ \ $

$ \displaystyle \ \ \ \ \ \ \ \ \ \ \ (\text{or)}$
$ \displaystyle \ \ \ \ \ 3\theta =360{}^\circ +180{}^\circ -45{}^\circ \ $

$ \displaystyle \ \,\ \ \ 3\theta =495{}^\circ \ $

$ \displaystyle \ \,\ \ \ \theta =165{}^\circ \ $

$ \displaystyle \ \ \ \ \ \ \ \ \ \ \ (\text{or)}\ $
$ \displaystyle \ \ \ \ \ 3\theta =360{}^\circ +360{}^\circ +180{}^\circ -45{}^\circ \ $

$ \displaystyle \ \,\ \ \ 3\theta =855{}^\circ \ $

$ \displaystyle \ \,\ \ \ \theta =285{}^\circ $

$ \displaystyle \underline{{\text{For }{{\text{4}}^{{\text{th}}}}\ \text{quadrant}}}\text{,}$

$ \displaystyle \ \,\ \ \ 3\theta =360{}^\circ -45{}^\circ $

$ \displaystyle \ \,\ \ \ 3\theta =315{}^\circ \ $

$ \displaystyle \ \,\ \ \ \theta =105{}^\circ \ $

$ \displaystyle \ \ \ \ \ \ \ \ \ \ \ (\text{or)}\ $
$ \displaystyle \ \ \ \ \ 3\theta =360{}^\circ +360{}^\circ -45{}^\circ $

$ \displaystyle \ \,\ \ \ 3\theta =675{}^\circ \ $

$ \displaystyle \ \,\ \ \ \theta =225{}^\circ \ \ $

$ \displaystyle \ \ \ \ \ \ \ \ \ \ \ (\text{or)}\ $
$ \displaystyle \ \ \ \ \ 3\theta =360{}^\circ +360{}^\circ +360{}^\circ -45{}^\circ \ $

$ \displaystyle \ \,\ \ \ 3\theta =1035{}^\circ \ $

$ \displaystyle \ \,\ \ \ \theta =345{}^\circ \ \ $

diagram မ်ားသည္ principal angle မ်ား ကို ဆံုးျဖတ္ရာတြင္  အေထာက္အကူ ျဖစ္ေစရန္ ေရးဆြဲ ေဖၚျပျခင္း ျဖစ္သည္။ နားလည္ ကၽြမ္းက်င္ သြားလ်င္ diagram မ်ား မေရးဆြဲပဲ ေျဖဆိုႏိုင္ ပါသည္။ က်န္ေသာပုစာၦမ်ားအတြက္ diagram မဆြဲေတာ့ပဲ ေျဖဆိုပါမည္။



(g) $ \displaystyle \ \ \ \ \ \ \tan (3\theta -30{}^\circ )=-1$

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$ \displaystyle \begin{array}{l}\therefore \ \ \ \ \text{basic acute angle}=45{}^\circ \\\\\ \ \ \ \ \ \text{But }\tan (3\theta -30{}^\circ )\ \text{is negative, }(3\theta -30{}^\circ )\ \text{may be }\\\ \ \ \ \ \ \text{either in the }{{\text{2}}^{{\text{nd}}}}\text{ or }{{\text{4}}^{{\text{th}}}}\text{ quadrant}\text{.}\\\\\underline{{\text{For }{{\text{2}}^{{\text{nd}}}}\ \text{quadrant}}}\text{,}\\\\\ \,\ \ \ 3\theta -30{}^\circ =180{}^\circ -45{}^\circ \\\ \\\ \,\ \ \ 3\theta =165{}^\circ \ \\\\\ \,\ \ \ \theta =55{}^\circ \ \\\\\ \ \ \ \ \ \ \ \ \ \ (\text{or)}\ \\\\\ \ \ \ \ 3\theta -30{}^\circ =360{}^\circ +180{}^\circ -45{}^\circ \ \\\\\ \,\ \ \ 3\theta =525{}^\circ \ \\\\\ \,\ \ \ \theta =175{}^\circ \ \\\\\ \ \ \ \ \ \ \ \ \ \ (\text{or)}\ \\\\\ \ \ \ \ 3\theta -30{}^\circ =360{}^\circ +360{}^\circ +180{}^\circ -45{}^\circ \ \\\\\ \,\ \ \ 3\theta =885{}^\circ \ \\\\\ \,\ \ \ \theta =295{}^\circ \ \ \ \\\\\underline{{\text{For }{{\text{4}}^{{\text{th}}}}\ \text{quadrant}}}\text{,}\\\\\ \,\ \ \ 3\theta -30{}^\circ =360{}^\circ -45{}^\circ \\\\\ \,\ \ \ 3\theta =345{}^\circ \ \\\\\ \,\ \ \ \theta =115{}^\circ \ \ \\\\\ \ \ \ \ \ \ \ \ \ \ (\text{or)}\ \\\\\ \ \ \ \ 3\theta -30{}^\circ =360{}^\circ +360{}^\circ -45{}^\circ \\\\\ \,\ \ \ 3\theta =705{}^\circ \ \\\\\ \,\ \ \ \theta =295{}^\circ \ \ \\\\\ \ \ \ \ \ \ \ \ \ \ (\text{or)}\ \\\\\ \ \ \ \ 3\theta -30{}^\circ =360{}^\circ +360{}^\circ +360{}^\circ -45{}^\circ \ \\\\\ \,\ \ \ 3\theta =1065{}^\circ \ \\\\\ \,\ \ \ \theta =355{}^\circ \ \ \end{array}$


(h) $ \displaystyle \ \ \ \ \ \ \cos 2\theta =-\frac{1}{2}$

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$ \displaystyle \begin{array}{l}\therefore \ \ \ \ \text{basic acute angle}=60{}^\circ \\\\\ \ \ \ \ \ \text{But }\cos 2\theta \ \text{is negative, }2\theta \ \text{may be }\\\ \ \ \ \ \ \text{either in the }{{\text{2}}^{{\text{nd}}}}\text{ or }{{\text{3}}^{{\text{rd}}}}\text{ quadrant}\text{.}\\\\\underline{{\text{For }{{\text{2}}^{{\text{nd}}}}\ \text{quadrant}}}\text{,}\\\\\ \,\ \ \ 2\theta =180{}^\circ -60{}^\circ \\\ \\\ \,\ \ \ 2\theta =120{}^\circ \ \\\\\ \,\ \ \ \theta =60{}^\circ \ \\\\\ \ \ \ \ \ \ \ \ \ \ (\text{or)}\ \\\\\ \ \ \ \ 2\theta =360{}^\circ +180{}^\circ -60{}^\circ \ \\\\\ \,\ \ \ 2\theta =480{}^\circ \ \\\\\ \,\ \ \ \theta =240{}^\circ \ \\\\\underline{{\text{For }{{\text{3}}^{{\text{rd}}}}\ \text{quadrant}}}\text{,}\\\\\ \,\ \ \ 2\theta =180{}^\circ +60{}^\circ \\\\\ \,\ \ \ 2\theta =240{}^\circ \ \\\\\ \,\ \ \ \theta =120{}^\circ \ \ \\\\\ \ \ \ \ \ \ \ \ \ \ (\text{or)}\ \\\\\ \ \ \ \ 2\theta =360{}^\circ +180{}^\circ +60{}^\circ \\\\\ \,\ \ \ 2\theta =600{}^\circ \ \\\\\ \,\ \ \ \theta =300{}^\circ \ \ \ \ \end{array}$


(i) $ \displaystyle \ \ \ \ \ \sin (2\theta +30{}^\circ )=\frac{{\sqrt{3}}}{2}$

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$ \displaystyle \begin{array}{l}\therefore \ \ \ \ \text{basic acute angle}=60{}^\circ \\\\\ \ \ \ \ \ \text{But }\sin (2\theta +30{}^\circ )\ \text{is positive, }2\theta +30{}^\circ \ \text{may be }\\\ \ \ \ \ \ \text{either in the }{{\text{1}}^{{\text{st}}}}\text{ or }{{\text{2}}^{{\text{nd}}}}\text{ quadrant}\text{.}\\\\\underline{{\text{For }{{\text{1}}^{{\text{st}}}}\ \text{quadrant}}}\text{,}\\\\\ \,\ \ \ 2\theta +30{}^\circ =60{}^\circ \\\ \\\ \,\ \ \ 2\theta =30{}^\circ \ \\\\\ \,\ \ \ \theta =15{}^\circ \ \\\\\ \ \ \ \ \ \ \ \ \ \ (\text{or)}\ \\\\\ \ \ \ \ 2\theta =360{}^\circ +60{}^\circ \ \\\\\ \,\ \ \ 2\theta =420{}^\circ \ \\\\\ \,\ \ \ \theta =210{}^\circ \ \\\\\underline{{\text{For }{{\text{2}}^{{\text{nd}}}}\ \text{quadrant}}}\text{,}\\\\\ \,\ \ \ 2\theta +30{}^\circ =180{}^\circ -60{}^\circ \\\ \\\ \,\ \ \ 2\theta =120{}^\circ \ \\\\\ \,\ \ \ \theta =60{}^\circ \ \\\\\ \ \ \ \ \ \ \ \ \ \ (\text{or)}\ \\\\\ \ \ \ \ 2\theta =360{}^\circ +180{}^\circ -60{}^\circ \ \\\\\ \,\ \ \ 2\theta =480{}^\circ \ \\\\\ \,\ \ \ \theta =240{}^\circ \ \end{array}$


(j) $ \displaystyle \ \ \ \ \ \ \tan \frac{1}{2}\theta =-\frac{1}{{\sqrt{3}}}$

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$ \displaystyle \therefore \ \ \ \ \text{basic acute angle}=30{}^\circ $

$ \displaystyle \ \ \ \ \ \ \text{But}\ \tan \frac{1}{2}\theta \ \text{is positive, }\frac{1}{2}\theta \ \text{may be }$

$ \displaystyle \ \ \ \ \ \ \text{either in the }{{\text{2}}^{{\text{nd}}}}\text{ or }{{\text{4}}^{{\text{th}}}}\text{ quadrant}\text{.}$

$ \displaystyle \therefore \ \ \ \ \frac{1}{2}\theta =180{}^\circ -30{}^\circ \ (\text{or})\ \frac{1}{2}\theta =360{}^\circ -30{}^\circ $

$ \displaystyle \therefore \ \ \ \ \frac{1}{2}\theta =150{}^\circ \ (\text{or})\ \frac{1}{2}\theta =330{}^\circ $

$ \displaystyle \therefore \ \ \ \ \theta =300{}^\circ \ (\text{or})\ \theta =660{}^\circ $

$ \displaystyle \begin{array}{l}\ \ \ \ \ \ \text{But}\ 0{}^\circ \le \theta \le 360{}^\circ ,\theta =660{}^\circ \ \text{is not in domain}\text{.}\\\\\therefore \ \ \ \ \theta =300{}^\circ \ \text{is the only solution}\text{.}\end{array}$

(k) $ \displaystyle \ \ \ \ \ \ \sin \theta =0.6521$

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$ \displaystyle \begin{array}{l}\therefore \ \ \ \ \text{basic acute angle}=40{}^\circ 4{2}'\ (\text{using table)}\\\\\ \ \ \ \ \ \text{But}\ \sin \theta \ \text{is positive, }\theta \ \text{may be }\\\ \ \ \ \ \ \text{either in the }{{\text{1}}^{{\text{st}}}}\text{ or }{{\text{2}}^{{\text{nd}}}}\text{ quadrant}\text{.}\\\\\therefore \ \ \ \ \theta =40{}^\circ 4{2}'\ (\text{or})\ \theta =180{}^\circ -40{}^\circ 4{2}'\\\\\therefore \ \ \ \ \theta =40{}^\circ 4{2}'\ (\text{or})\ \theta =139{}^\circ 1{8}'\\\ \end{array}$


(l) $ \displaystyle \ \ \ \ \ \ \cos \theta =-0.3854$

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$ \displaystyle \begin{array}{l}\therefore \ \ \ \ \text{basic acute angle}=67{}^\circ 2{0}'\ (\text{using table)}\\\\\ \ \ \ \ \ \text{But}\ \cos \theta \ \text{is negative, }\theta \ \text{may be }\\\ \ \ \ \ \ \text{either in the }{{\text{2}}^{{\text{nd}}}}\text{ or }{{\text{3}}^{{\text{rd}}}}\text{ quadrant}\text{.}\\\\\therefore \ \ \ \ \theta =180{}^\circ -67{}^\circ 2{0}'\ (\text{or})\ \theta =180{}^\circ +67{}^\circ 2{0}'\\\\\therefore \ \ \ \ \theta =112{}^\circ 4{0}'\ (\text{or})\ \theta =247{}^\circ 2{0}'\end{array}$