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三角比率倒数

学习cosecant, secant,和cotangent是基本三角函数:sine, cosine, and tangent.的倒数。
我们已经学了基础的锐角三角比:
但还有另外三个比例要学习:
  • 基于 start fraction, start color #11accd, a, end color #11accd, divided by, start color #aa87ff, c, end color #aa87ff, end fraction, 我们可以得到 start fraction, start color #aa87ff, c, end color #aa87ff, divided by, start color #11accd, a, end color #11accd, end fraction.
  • 基于start fraction, start color #ed5fa6, b, end color #ed5fa6, divided by, start color #aa87ff, c, end color #aa87ff, end fraction,我们可以得到 start fraction, start color #aa87ff, c, end color #aa87ff, divided by, start color #ed5fa6, b, end color #ed5fa6, end fraction.
  • 基于 start fraction, start color #11accd, a, end color #11accd, divided by, start color #ed5fa6, b, end color #ed5fa6, end fraction, 我们可以得到 start fraction, start color #ed5fa6, b, end color #ed5fa6, divided by, start color #11accd, a, end color #11accd, end fraction.
这些新的比例是 反三角函数, 我们即将学到这些比例的名称。

余割 left parenthesis, \csc, right parenthesis

余割 是正弦的倒数。它是直角三角形的斜边与特定角所对边的比值 。
sine, left parenthesis, A, right parenthesis, equals, start fraction, start color #11accd, start text, 对, 边, end text, end color #11accd, divided by, start color #aa87ff, start text, 斜, 边, end text, end color #aa87ff, end fraction, equals, start fraction, start color #11accd, a, end color #11accd, divided by, start color #aa87ff, c, end color #aa87ff, end fraction
\csc, left parenthesis, A, right parenthesis, equals, start fraction, start color #aa87ff, start text, 斜, 边, end text, end color #aa87ff, divided by, start color #11accd, start text, 对, 边, end text, end color #11accd, end fraction, equals, start fraction, start color #aa87ff, c, end color #aa87ff, divided by, start color #11accd, a, end color #11accd, end fraction

割线left parenthesis, \sec, right parenthesis

割线是余弦的倒数。它是直角三角形与特定角的邻边的比值。
cosine, left parenthesis, A, right parenthesis, equals, start fraction, start color #ed5fa6, start text, 邻, 边, end text, end color #ed5fa6, divided by, start color #aa87ff, start text, 对, 边, end text, end color #aa87ff, end fraction, equals, start fraction, start color #ed5fa6, b, end color #ed5fa6, divided by, start color #aa87ff, c, end color #aa87ff, end fraction
\sec, left parenthesis, A, right parenthesis, equals, start fraction, start color #aa87ff, start text, 斜, 边, end text, end color #aa87ff, divided by, start color #ed5fa6, start text, 对, 边, end text, end color #ed5fa6, end fraction, equals, start fraction, start color #aa87ff, c, end color #aa87ff, divided by, start color #ed5fa6, b, end color #ed5fa6, end fraction

余切 left parenthesis, cotangent, right parenthesis

余切是正切的倒数。 它是直角三角形中邻边与对边的比值。
tangent, left parenthesis, A, right parenthesis, equals, start fraction, start color #11accd, start text, 对, 边, end text, end color #11accd, divided by, start color #ed5fa6, start text, 邻, 边, end text, end color #ed5fa6, end fraction, equals, start fraction, start color #11accd, a, end color #11accd, divided by, start color #ed5fa6, b, end color #ed5fa6, end fraction
cotangent, left parenthesis, A, right parenthesis, equals, start fraction, start color #ed5fa6, start text, 邻, 边, end text, end color #ed5fa6, divided by, start color #11accd, start text, 对, 边, end text, end color #11accd, end fraction, equals, start fraction, start color #ed5fa6, b, end color #ed5fa6, divided by, start color #11accd, a, end color #11accd, end fraction

人们如何记住这些?

对大部分来说,通过将新的比例和他们的倒数相关联是比较容易记忆的。 下表总结了这种关系。
文字描述数学关系
cosecantcosecantsine的倒数\csc, left parenthesis, A, right parenthesis, equals, start fraction, 1, divided by, sine, left parenthesis, A, right parenthesis, end fraction
secantsecantcosine的倒数\sec, left parenthesis, A, right parenthesis, equals, start fraction, 1, divided by, cosine, left parenthesis, A, right parenthesis, end fraction
cotangentcotangenttangent的倒数cotangent, left parenthesis, A, right parenthesis, equals, start fraction, 1, divided by, tangent, left parenthesis, A, right parenthesis, end fraction

找到反三角函数

我们来看个例子。

下图三角形中,求 \csc, left parenthesis, C, right parenthesis, \sec, left parenthesis, C, right parenthesis, and cotangent, left parenthesis, C, right parenthesis.

解题

求 cosecant (余割)
我们知道 cscsin的倒数
因为 sine 是对边比斜边,cosecant 就是斜边比对边。
csc(C)=斜边 对边=1715\begin{aligned}\csc (C) &= \dfrac{\purpleC{\text{斜边}}} {\blueD{\text{ 对边}}} \\\\ &= \dfrac{{17}}{{15}} \end{aligned}
求 secant (sec)
我们知道 seccos的倒数
因为 cos是邻边比斜边, sec就是斜边比邻边。
sec(C)=对边邻边=178\begin{aligned}\sec (C) &= \dfrac{\purpleC{\text{对边}}}{\maroonC{\text{邻边}}} \\\\ &= \dfrac{{17}}{{8}} \end{aligned}
求 cotangent (余切)
我们知道cottan的倒数
因为 tan是对边比邻边, cot 就是邻边比对边
cot(C)=邻边对边=815\begin{aligned}\cot (C) &= \dfrac{\maroonC{\text{邻边}}}{\blueD{\text{对边}}} \\\\ &= \dfrac{{8}}{{15}} \end{aligned}

自己试一试!

问题1
\csc, left parenthesis, X, right parenthesis, equals
  • 你的答案是
  • 一个最简真分数,如 3, slash, 5
  • 一个最简假分数,如 7, slash, 4

问题2
\sec, left parenthesis, W, right parenthesis, equals
  • 你的答案是
  • 一个最简真分数,如 3, slash, 5
  • 一个最简假分数,如 7, slash, 4

问题3
cotangent, left parenthesis, R, right parenthesis, equals
  • 你的答案是
  • 一个最简真分数,如 3, slash, 5
  • 一个最简假分数,如 7, slash, 4

挑战题
\csc, left parenthesis, 45, degrees, right parenthesis的值是多少?