【数学I】例題4.1.13：三角比を含む不等式3（One More）★★★

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% 例題I4.1.13：三角比を含む不等式3（One More）★★★
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次の不等式を解け．ただし，$0^{\circ}\leqq\theta\leqq 180^{\circ}$とする．

(1) $2\cos^2\theta+\cos\theta-1\geqq 0$

(2) $8\cos^2\theta<1+10\sin\theta$

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% 例題I4.1.13の解答（One More）★★★
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(1) $2\cos^2\theta+\cos\theta-1\geqq 0\cdots(\mathrm{i})$とする．

$\cos\theta=t$とおくと，$0^{\circ}\leqq\theta\leqq 180^{\circ}$より，$-1\leqq t\leqq 1\cdots(\mathrm{ii})$

また，(i)の不等式は，$2t^2+t-1\geqq 0$より，$(t+1)(2t-1)\geqq 0$

したがって，$t\leqq-1,\frac{1}{2}\leqq t$

これと(ii)より，$t=-1,\frac{1}{2}\leqq t\leqq 1$

すなわち，$\cos\theta=-1,\frac{1}{2}\leqq\cos\theta\leqq 1$

よって，求める解は，$0^{\circ}\leqq\theta\leqq 60^{\circ},\theta=180^{\circ}$

(2) $8\cos^2\theta<1+10\sin\theta$より，$8(1-\sin^2\theta)<1+10\sin\theta$

したがって，$8\sin^2\theta+10\sin\theta-7>0\cdots(\mathrm{i})$

$\sin\theta=t$とおくと，$0^{\circ}\leqq\theta\leqq 180^{\circ}$より，$0\leqq t\leqq 1\cdots(\mathrm{ii})$

また，(i)の不等式は，$8t^2+10t-7>0$より，$(4t+7)(2t-1)>0$

ゆえに，$t<-\frac{7}{4},\frac{1}{2}<t$

これと(ii)より，$\frac{1}{2}<t\leqq 1$

すなわち，$\frac{1}{2}<\sin\theta\leqq 1$

よって，求める解は，$30^{\circ}<\theta<150^{\circ}$

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% 問題I4.1.13：三角比を含む不等式3（One More）★★★
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次の不等式を解け．ただし，$0^{\circ}\leqq\theta\leqq 180^{\circ}$とする．

(1) $2\sin^2\theta+\cos\theta-2>0$

(2) $2\sin^2\theta-3\sin\theta+1\geqq 0$

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% 問題I4.1.13の解答（One More）★★★
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(1) $2\sin^2\theta+\cos\theta-2>0$より，$2(1-\cos^2\theta)+\cos\theta-2>0$

したがって，$2\cos^2\theta-\cos\theta<0\cdots(\mathrm{i})$

$\cos\theta=t$とおくと，$0^{\circ}\leqq\theta\leqq 180^{\circ}$より，$-1\leqq t\leqq 1\cdots(\mathrm{ii})$

また，(i)の不等式は，$2t^2-t<0$より，$t(2t-1)<0$

ゆえに，$0<t<\frac{1}{2}$

これと(ii)より，$0<\cos\theta<\frac{1}{2}$

よって，求める解は，$60^{\circ}<\theta<90^{\circ}$

(2) $2\sin^2\theta-3\sin\theta+1\geqq 0$より，$2t^2-3t+1\geqq 0\cdots(\mathrm{i})$

$\sin\theta=t$とおくと，$0^{\circ}\leqq\theta\leqq 180^{\circ}$より，$0\leqq t\leqq 1\cdots(\mathrm{ii})$

また，(i)の不等式は，$2t^2-3t+1\geqq 0$より，$(2t-1)(t-1)\geqq 0$

したがって，$t\leqq\frac{1}{2},1\leqq t$

これと(ii)より，$0\leqq t\leqq\frac{1}{2},t=1$

すなわち，$0\leqq\sin\theta\leqq\frac{1}{2},\sin\theta=1$

よって，求める解は，$0^{\circ}\leqq\theta\leqq 30^{\circ},150^{\circ}\leqq\theta\leqq 180^{\circ},\theta=90^{\circ}$

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% 例題I4.1.13の側注：三角比を含む不等式3（One More）★★★
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% 側注1
見やすさと扱いやすさから，おき換えることを推奨しているが，おき換えずに考えてもよい．

% 側注2
$\sin^2\theta+\cos^2\theta=1$

% 側注3
たすき掛けを用いる．

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% 問題I4.1.13の側注：三角比を含む不等式3（One More）★★★
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% 側注1
$\sin^2\theta+\cos^2\theta=1$

% 側注2
たすき掛けを用いる．

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% 例題I4.1.13の考え方・補足など：三角比を含む不等式3（One More）★★★
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$0^{\circ}\leqq\theta\leqq 180^{\circ}$のとき，$-1\leqq\cos\theta\leqq 1,0\leqq\sin\theta\leqq 1$であることに注意して，不等式を解く．(2)は，$\sin^2\theta+\cos^2\theta=1$，すなわち，$\cos^2\theta=1-\sin^2\theta$を利用して，三角比の種類を減らす．

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三角比を含む方程式・不等式は三角比の種類を減らすとよい．

また，$0^{\circ}\leqq\theta\leqq 180^{\circ}$のとき，$-1\leqq\cos\theta\leqq 1,0\leqq\sin\theta\leqq 1$であることに注意すること．