2020年03月30日

高校数学「三角関数」「三角不等式」「4(sinθ)^2−1>0」

高校数学「三角関数」「三角不等式」「4(sinθ)^2−1>0」

■問題

4(sinθ)^2−1>0を解け。ただし、0≦θ<2πとする。


三角不等式のちょっとだけ難しい問題です。このくらいが解ければ、定期テストでもそこそこ解けると思います。


↓三角不等式などの三角関数の問題の解き方がマスターできるテキストです↓



■解答解説

まずはsinθの2次式ととらえて、sinθの範囲を出すことを考えます。
t=sinθとおくと解きやすいですね。
(慣れている人はそのままでもOKです)

t=sinθとすると、

4t^2−1>0

これはただの2次不等式ですね。普通に解いてみましょう!

(2t+1)(2t−1)>0

よって、t<−1/2,t>1/2

t=sinθより、sinθ<−1/2,sinθ>1/2

これらの式を満たすθの範囲が求める範囲となります。つまり、

sinθ=−1/2のときθ=(7/6)π,(11/6)πだから、(7/6)π<θ<(11/6)π
sinθ=1/2のときθ=π/6,(5/6)πだから、π/6<θ<(5/6)π


関連問題
2次不等式


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posted by えま at 21:51| Comment(0) | 高校数学 | このブログの読者になる | 更新情報をチェックする

本日配信のメルマガ。2020年センター英語第5問 本文序盤

本日配信のメルマガでは、2020年大学入試センター試験英語第5問の本文序盤のスラッシュリーディングを掲載します。


【高校英語】過去問攻略!センター英語
http://www.mag2.com/m/0001641009.html


■ 問題

第5問 次の文章を読み、下の問い(問1〜5)の[ 41 ]〜[ 45 ]に入れる
のに最も適当なものを、それぞれ下の{1}〜{4}のうちから一つずつ選べ。

A couple of weeks ago, I was hiking with my dog on a mountain when
something unexpected happened and I lost sight of him. I looked and looked
but couldn't find him. He had been with me for so long that it was like I
was missing part of my soul.
Ever since that day, I had a strange feeling. It was beyond sadness --
a feeling that I didn't quite understand, as if something were pulling me
to go back to the mountain. So every chance I got, I grabbed my backpack
to see if the mountain could offer me some sense of relief.
One sunny morning, I stood at the foot of the mountain. Something felt
different this day. "Please forgive me," I said out loud. "I'll find you!"
I took a deep breath and began my journey with this mysterious pull growing
stronger. After making my way along paths I thought I knew well, I realized
I was somehow in an unfamiliar place. I panicked a little, lost my footing,
and slipped. From out of nowhere, an elderly man came running towards me
and helped me up.
Looking at his gentle, smiling face, I felt a sense of ease. The old man
said he was looking for a way to the top of the mountain, so we decided to
climb together.
Soon the path began to feel familiar again. We talked about many things,
including my dog. I told him that he was a German shepherd. When he was
younger, he served briefly as a police dog but had to stop due to an
injury. The man let out a laugh saying he had been a police officer for
a short time, but he quit. He didn't say why. Later, he spent a long time
as a bodyguard. He also had German roots. We laughed at these similarities.
Before we knew it, we reached a large open area and took a break. I told
the man what had happened to my dog. "He had a tiny bell on his collar to
scare away bears. We came to this very spot and saw a bear. It was looking
back at us. I should have held my dog because, sensing danger, he chased
after the bear. I couldn't find him after that. I should have been more
careful."
As I was telling the story, the man's expression changed. "It wasn't your
fault. Your dog just wanted to keep you safe," he said. "I'm sure Tomo
would want to tell you this. Also, thank you for not giving up."
Tomo is my dog's name. Did I tell him this? The old man's comment [rang
in the air].
Before I could ask anything, the man proposed we hurry to get to the top
of the mountain. I was planning to do this with my dog a few weeks ago.
After two more hours of hiking, we reached the peak. I set down my backpack
and we sat taking in the magnificent view. The old man looked at me and
said, "Mountains offer truly magical experiences."
I looked around for a place to rest. I guess I was pretty tired, because
I fell asleep right away. When I woke up, I noticed that the old many had
disappeared. I waited, but he never returned.
Suddenly, in the sunlight, something caught my eye. I walked over and saw
a small metal tag beside my backpack. It was the same silver name tag that
my parents originally gave to my dog. Tomo it said.
It was then that I heard a familiar noise behind me. It was the ringing
of a tiny bell. I turned around. When I saw caused so many emotions to rush
over me.
After a while on the mountaintop, I attached the name tag to my old friend
and carefully made my way home with the mountain's gift beside me. My soul
felt very much complete.


問1 The author kept returning the mountain because [ 41 ].
{1} she felt an urge she couldn't explain
{2} she planned to meet the elderly man
{3} she thought she could practice magic
{4] she wanted to find out about the bear

問2 Which of the following happened first on the author's most recent
trip? [ 42 ]
{1} She arrived at a large open area.
{2} She climbed to the mountaintop.
{3} She saw a bear running away.
{4} She was assisted by an old man.

問3 What similarity between the authour's dog and the old man was talked
about? [ 43 ]
{1} They experienced workplace injuries.
{2} They recently lost close family friends.
{3} They were acquaintances of the author.
{4} They worked to help protect the public.

問4 Which of the following is closest to the meaning of the underlined
phrase [rang in the air] as used in the text? [ 44 ]
{1} brought happiness
{2} left an impression
{3} make a loud noise
{4} seemed offensive

問5 How did the author's feelings change over the course of the last
hiking experience? [ 45 ]
{1} She was depressed and then became sadder.
{2} She was determined and then became comforted.
{3} She was hopeful but then became homesick.
{4} She was miserable but then became entertained.


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■ 解答・解説

A couple of weeks ago, / I was hiking / with my dog/ on a mountain
/ when / something unexpected happened/ and / I lost sight / of him.
2,3週間前 / 私はハイキングしていた / 犬と / ある山で
/ 〜のとき / 予想しないことが起きた / そして / 見失った / 彼を

I looked and looked / but / couldn't find him.
私は見て見た / しかし / 彼を見つけられなかった


(以下略)


(有料版では、解説の続きや語句コーナー、解答一覧も掲載しています)
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posted by えま at 17:29| Comment(0) | メルマガ | このブログの読者になる | 更新情報をチェックする
こんなヤツです
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