【数学I】例題4.1.12：三角比を含む不等式2（One More）★★

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% 例題I4.1.12：三角比を含む不等式2（One More）★★
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次の不等式を満たす$\theta$の値の範囲を求めよ．ただし，$0^{\circ}\leqq\theta\leqq 180^{\circ}$とする．

(1) $\tan\theta\geqq\sqrt{3}$

(2) $\tan\theta<1$

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% 例題I4.1.12の解答（One More）★★
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(1) $\tan\theta=\sqrt{3}$より，$\theta=60^{\circ}$

よって，右の図より，$\tan\theta\geqq\sqrt{3}$を満たす$\theta$の値の範囲は，

\[
60^{\circ}\leqq\theta<90^{\circ}
\]

(2) $\tan\theta=1$より，$\theta=45^{\circ}$

よって，右の図より，$\tan\theta<1$を満たす$\theta$の値の範囲は，

\[
0^{\circ}\leqq\theta<45^{\circ},90^{\circ}<\theta\leqq 180^{\circ}
\]

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% 問題I4.1.12：三角比を含む不等式2（One More）★★
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次の不等式を満たす$\theta$の値の範囲を求めよ．ただし，$0^{\circ}\leqq\theta\leqq 180^{\circ}$とする．

(1) $\tan\theta\leqq-\frac{1}{\sqrt{3}}$

(2) $\tan\theta>-1$

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% 問題I4.1.12の解答（One More）★★
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(1) $\tan\theta=-\frac{1}{\sqrt{3}}$より，$\theta=150^{\circ}$

よって，右の図より，$\tan\theta\leqq-\frac{1}{\sqrt{3}}$の値の範囲は，

\[
90^{\circ}<\theta\leqq 150^{\circ}
\]

(2) $\tan\theta=-1$より，$\theta=135^{\circ}$

よって，右の図より，$\tan\theta>-1$の値の範囲は，

\[
0^{\circ}\leqq\theta<90^{\circ},135^{\circ}<\theta\leqq 180^{\circ}
\]

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% 例題I4.1.12の側注：三角比を含む不等式2（One More）★★
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% 側注1
不等号を等号におき換える．

% 側注2
傾きが$\sqrt{3}$以上となるような範囲を求める．

% 側注3
$\theta\neq 90^\circ$であることに注意すること．

% 側注4
不等号を等号におき換える．

% 側注5
傾きが$1$より小さい範囲を求める．

% 側注6
$\theta\neq 90^\circ$であることに注意すること．

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% 問題I4.1.12の側注：三角比を含む不等式2（One More）★★
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% 側注1
不等号を等号におき換える．

% 側注2
傾きが$-\frac{1}{\sqrt{3}}$以下となる範囲を求める．

% 側注3
$\theta\neq 90^\circ$であることに注意すること．

% 側注4
不等号を等号におき換える．

% 側注5
傾きが$-1$より大きい範囲を求める．

% 側注6
$\theta\neq 90^\circ$であることに注意すること．

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% 例題I4.1.12の考え方・補足など：三角比を含む不等式2（One More）★★
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% 考え方
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不等式を解くために，不等号を等号におき換える．

$\tan\theta=k$のとき，$\tan\theta$は$y=kx$の傾きであるから，右の図のように，直線$y=kx$と直線$x=1$を考える．半径1の半円との関係に注目し，$\tan\theta>k$のとき，傾きが$k$より大きい範囲，$\tan\theta<k$のとき，傾き$k$より小さい範囲を求める．そして，$k$の範囲に対応する，$\theta$の範囲を求めればよい．このとき，$\theta\neq 90^\circ$であることに注意すること．

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% 注意
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$\tan\theta$は$90^\circ$で定義されないので，$\theta$の範囲は$0^{\circ}\leqq\theta<90^{\circ}$と$90^{\circ}<\theta\leqq 180^{\circ}$に分けて考えることに注意すること．