【数学I】例題4.1.11：三角比を含む不等式1（One More）★★

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% 例題I4.1.11：三角比を含む不等式1（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) $\sin\theta\geqq\frac{1}{2}$

(2) $\cos\theta<-\frac{1}{2}$

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% 例題I4.1.11の解答（One More）★★
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(1) $\sin\theta=\frac{1}{2}$より，$\theta=30^{\circ},150^{\circ}$

よって，右の図より，$\sin\theta\geqq\frac{1}{2}$となる$\theta$の値の範囲は，

\[
30^{\circ}\leqq\theta\leqq 150^{\circ}
\]

(2) $\cos\theta=-\frac{1}{2}$より，$\theta=120^{\circ}$

よって，右の図より，$\cos\theta<-\frac{1}{2}$となる$\theta$の値の範囲は，

\[
120^{\circ}<\theta\leqq 180^{\circ}
\]

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% 問題I4.1.11：三角比を含む不等式1（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) $\sin\theta>\frac{\sqrt{3}}{2}$

(2) $\cos\theta\leqq-\frac{\sqrt{2}}{2}$

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% 問題I4.1.11の解答（One More）★★
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(1) $\sin\theta=\frac{\sqrt{3}}{2}$より，$\theta=60^{\circ},120^{\circ}$

よって，右の図より，$\sin\theta>\frac{\sqrt{3}}{2}$となる$\theta$の値の範囲は，

\[
60^{\circ}<\theta<120^{\circ}
\]

(2) $\cos\theta=-\frac{\sqrt{2}}{2}$より，$\theta=135^{\circ}$

よって，右の図より，$\cos\theta\leqq-\frac{\sqrt{2}}{2}$となる$\theta$の値の範囲は，

\[
135^{\circ}\leqq\theta\leqq 180^{\circ}
\]

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% 例題I4.1.11の側注：三角比を含む不等式1（One More）★★
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% 側注1
不等号を等号におき換えて，方程式を解く．

% 側注2
等号を含むか否かに注意すること．

% 側注3
不等号を等号におき換えて，方程式を解く．

% 側注4
等号を含むか否かに注意すること．

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% 問題I4.1.11の側注：三角比を含む不等式1（One More）★★
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% 側注1
不等号を等号におき換えて，方程式を解く．

% 側注2
等号を含むか否かに注意すること．

% 側注3
不等号を等号におき換えて，方程式を解く．

% 側注4
等号を含むか否かに注意すること．

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% 例題I4.1.11の考え方・補足など：三角比を含む不等式1（One More）★★
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% 考え方
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三角比を含む不等式を解くときは，不等号を等号におき換えて，三角比を含む方程式を解くことを考えるとよい．方程式の解から，半径1の半円を用いて不等式を満たす$\theta$の範囲を求めることができる．

(1) $\sin\theta=k$のとき，$\sin\theta$は半円上の$y$座標であるから，右の図のように，直線$y=k$を考える．半円との関係に注目すると，半円上の点で$\sin\theta\geqq k$であるから，$y=k$より上側である$y\geqq k$となる$\theta$の範囲を求めればよい．

(1)

(2) $\cos\theta=k$のとき，$\cos\theta$は半円上の$x$座標であるから，右の図のように，直線$x=k$を考える．半円との関係に注目すると，半円上の点で$\cos\theta<k$であるから，$x=k$より左側である$x<k$となる$\theta$の範囲を求めればよい．

(2)

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三角比を含む不等式は，半径1の半円をかき，大小関係を考える．