Course: 8.10 逆三角比 | The Way To Learn

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逆三角比
Inverse Trigonometric Ratios
::逆三角比

In mathematics, the word inverse means “undo.” For example, addition and subtraction are inverses of each other because one undoes the other. When we use the inverse trigonometric ratios, we can find acute angle measures as long as we are given two sides.
::在数学中,倒数字意为“不做 ” 。比如,增减是相互反差的,因为两者相互反差。当我们使用逆三角比时,只要给两边,我们就能发现急性角测量法。

Inverse Tangent: Labeled $\begin{align*}\tan^{-1}\end{align*}$ , the “-1” means inverse.
::逆切数: 标签的 tan-1, “-1” 表示反切数。

$\begin{align*}\tan^{-1} \left (\frac{b}{a} \right ) = m \angle B\end{align*}$ and $\begin{align*} \tan^{-1} \left( \frac{a}{b} \right) = m \angle A\end{align*}$ .
::-1(ba)=mB和tan_1(ab)=mA。

Inverse Sine: Labeled $\begin{align*}\sin^{-1}\end{align*}$ .
::倒弦: 标签的罪孽-1。

$\begin{align*}\sin^{-1} \left( \frac{b}{c} \right) = m \angle B\end{align*}$ and $\begin{align*} \sin^{-1} \left(\frac{a}{c} \right) = m \angle A\end{align*}$ .
:bc)=mB和sin_1(ac)=mA。

Inverse Cosine: Labeled $\begin{align*}\cos^{-1}\end{align*}$ .
::逆余弦: 标签COS-1 1 。

$\begin{align*}\cos^{-1} \left(\frac{a}{c} \right) = m \angle B\end{align*}$ and $\begin{align*} \cos^{-1} \left( \frac{b}{c} \right) = m \angle A\end{align*}$ .
::COS-1(ac)=mB和cos-1(bc)=mA。

In most problems, to find the measure of the angles you will need to use your calculator. On most scientific and graphing calculators, the buttons look like $\begin{align*}[ \text{SIN}^{-1}], [ \text{COS}^{-1}]\end{align*}$ , and $\begin{align*}[\text{TAN}^{-1}]\end{align*}$ . You might also have to hit a shift or $\begin{align*}2^{nd}\end{align*}$ button to access these functions.
::在大多数问题中, 要找到您需要使用计算器来计算角度的度量。在大多数科学和图形计算器中, 按钮看上去像 [SIN-1] 、 [COS-1] 和 [TAN-1] 。您可能还要按一个转换或第二个按钮才能访问这些函数。

Now that you know both the trig ratios and the inverse trig ratios you can solve a right triangle. To solve a right triangle, you need to find all sides and angles in it. You will usually use sine, cosine, or tangent; inverse sine, inverse cosine, or inverse tangent; or the .
::既然您知道三角比和反三角比,您就可以解决右三角。要解决右三角,您需要找到右三角。要解决右三角,您需要找到右三角的所有边和角度。通常使用正弦、正弦或正弦;反弦、逆弦或反正弦;或反正切。

What if you were told the tangent of $\begin{align*}\angle Z\end{align*}$ is 0.6494? How could you find the measure of $\begin{align*}\angle Z\end{align*}$ ?
::狦禗0.69494盾 或тm2

Examples
::实例

Example 1
::例1

Solve the right triangle.
::解决右三角。

The two acute angles are congruent, making them both $\begin{align*}45^\circ\end{align*}$ . This is a 45-45-90 triangle. You can use the trigonometric ratios or the special right triangle ratios.
::这两个急性角度是相同的, 使它们都达到45。这是45- 45- 90三角形。您可以使用三角比或特殊右三角比。

Trigonometric Ratios
::三角比

$\begin{aligned} \tan 45^{\circ} & = \frac{15}{B C} & \sin 45^{\circ} = \frac{15}{A C} \\ B C & = \frac{15}{\tan 45^{\circ}} = 15 & A C = \frac{15}{\sin 45^{\circ}} \approx 21.21 \end{aligned}$
$\begin{align*}\tan 45^\circ &= \frac{15}{BC} && \sin 45^\circ = \frac{15}{AC}\\ BC &= \frac{15}{\tan 45^\circ} = 15 && \quad \ AC = \frac{15}{\sin 45^\circ} \approx 21.21\end{align*}$
::15 ACBC=15tan4515ACC=15sin4521.21

45-45-90 Triangle Ratios
::45-45-90 三角比率

$\begin{aligned} B C = A B = 15, A C = 15 \sqrt{2} \approx 21.21 \end{aligned}$
$\begin{align*}BC = AB = 15, AC = 15 \sqrt{2} \approx 21.21\end{align*}$
::BC=AB=15,AC=152 21.21

Example 2
::例2

Use the sides of the triangle and your calculator to find the value of $\begin{align*}\angle A\end{align*}$ . Round your answer to the nearest tenth of a degree .
::使用三角形的两边和您的计算器来找到 A 的值。将您的答复旋转到一个度的十分之一。

In reference to $\begin{align*}\angle A\end{align*}$ , we are given the opposite leg and the adjacent leg. This means we should use the tangent ratio.
::参照A, 我们得到的是相反的腿和相邻的腿, 这意味着我们应该使用正切比例。

$\begin{align*}\tan A = \frac{20}{25} = \frac{4}{5}\end{align*}$ . So, $\begin{align*}\tan^{-1} \frac{4}{5} = m \angle A\end{align*}$ . Now, use your calculator.
::TANA=2025=45。所以,TAN -145=mA。现在,使用您的计算器。

If you are using a TI-83 or 84, the keystrokes would be: $\begin{align*}[2^{nd}]\end{align*}$ [TAN] $\begin{align*}\left( \frac{4}{5} \right)\end{align*}$ [ENTER] and the screen looks like:
::如果您使用TI-83或84, 键盘的键盘是: [2nd][TAN](45)[ENTEER], 屏幕看起来像:

$\begin{aligned} m ∠ A \approx {38.7}^{\circ} \end{aligned}$
$\begin{align*}m \angle A \approx 38.7^\circ\end{align*}$
::38.7

Example 3
::例3

$\begin{align*}\angle A\end{align*}$ is an acute angle in a right triangle. Find $\begin{align*}m \angle A\end{align*}$ to the nearest tenth of a degree for $\begin{align*}\sin A = 0.68\end{align*}$ , $\begin{align*}\cos A = 0.85\end{align*}$ , and $\begin{align*}\tan A = 0.34\end{align*}$ .
::A 是右三角形中的急性角。找到 mA 至最接近的十分之一度的 sinA=0. 68, cosA=0. 85, tanA=0. 34 。

$\begin{align*}m \angle A = \sin^{-1} 0.68 \approx 42.8^\circ\end{align*}$
::=======================================================================================================================================================

$\begin{align*}m \angle A = \cos^{-1} 0.85 \approx 31.8^\circ\end{align*}$
::==================================================================================================================================

$\begin{align*}m \angle A = \tan^{-1} 0.34 \approx 18.8^\circ\end{align*}$
::18.8

Example 4
::例4

Solve the right triangle.
::解决右三角。

To solve this right triangle, we need to find $\begin{align*}AB, m \angle C\end{align*}$ and $\begin{align*}m \angle B\end{align*}$ . Use only the values you are given.
::要解析右三角形, 我们需要找到 AB、 mC 和 mB。只使用给定的值。

$\begin{align*}\underline{AB}\end{align*}$ : Use the Pythagorean Theorem.
::AB_:使用毕达哥里语定理。

$\begin{aligned} 24^{2} + A B^{2} & = 30^{2} \\ 576 + A B^{2} & = 900 \\ A B^{2} & = 324 \\ A B & = \sqrt{324} = 18 \end{aligned}$
$\begin{align*}24^2 + AB^2 &= 30^2\\ 576 + AB^2 &= 900\\ AB^2 &= 324\\ AB &= \sqrt{324} = 18\end{align*}$
::242+AB2=302576+AB2=900AB2=324AB=324=18

$\begin{align*}\underline{m \angle B}\end{align*}$ : Use the inverse sine ratio.
::mB_: 使用逆正弦比例。

$\begin{aligned} \sin B & = \frac{24}{30} = \frac{4}{5} \\ \sin^{- 1} (\frac{4}{5}) \approx {53.1}^{\circ} = m ∠ B \end{aligned}$
$\begin{align*}\sin B &= \frac{24}{30} = \frac{4}{5}\\ \sin^{-1} \left( \frac{4}{5} \right) \approx 53.1^\circ = m \angle B\end{align*}$
::B=2430=45sin -1 (45) 53.1mB

$\begin{align*}\underline{m \angle C}\end{align*}$ : Use the inverse cosine ratio.
::mC_:使用逆余弦比。

$\begin{aligned} \cos C = \frac{24}{30} = \frac{4}{5} ⟶ \cos^{- 1} (\frac{4}{5}) \approx {36.9}^{\circ} = m ∠ C \end{aligned}$
$\begin{align*}\cos C = \frac{24}{30} = \frac{4}{5} \longrightarrow \cos^{-1}\left( \frac{4}{5} \right) \approx 36.9^\circ = m \angle C\end{align*}$
::C=2430=45cos -1(45)36.9mC

Example 5
::例5

When would you use $\begin{align*}\sin\end{align*}$ and when would you use $\begin{align*}\sin^{-1}\end{align*}$ ?
::你什么时候用罪什么时候用罪什么时候用罪

You would use $\begin{align*}\sin\end{align*}$ when you are given an angle and you are solving for a missing side. You would use $\begin{align*}\sin^{-1}\end{align*}$ when you are given sides and you are solving for a missing angle.
::当你被赋予一个角度,而你正在解决一个缺失的一面的时候,你会利用罪恶。当你被赋予一面,而你正在解决一个缺失的一面的时候,你会使用罪恶。

Review
::回顾

Use your calculator to find $\begin{align*}m \angle A\end{align*}$ to the nearest tenth of a degree.
::使用您的计算器查找 maA 到一个度的十分之一。

Let $\begin{align*}\angle A\end{align*}$ be an acute angle in a right triangle. Find $\begin{align*}m \angle A\end{align*}$ to the nearest tenth of a degree.
::在右三角形中, 将 A 变成急性角。找到 mA 至 10 度的最接近的 10 度。

$\begin{align*}\sin A = 0.5684\end{align*}$
::A=0.5684

$\begin{align*}\cos A = 0.1234\end{align*}$
::COsA=0.1234

$\begin{align*}\tan A = 2.78\end{align*}$
::泰纳=2.78

$\begin{align*}\cos^{-1} 0.9845\end{align*}$
::COS - 10.9845

$\begin{align*}\tan^{-1} 15.93\end{align*}$
::- 坦 - 坦 - 115.93

$\begin{align*}\sin^{-1} 0.7851\end{align*}$
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Solving the following right triangles. Find all missing sides and angles. Round any decimal answers to the nearest tenth.
::解决以下右三角形。查找所有缺失的侧面和角度。将小数点后的任何答案舍入到最近的十分点。

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

Resources
::资源

Section outline

General

逆三角比

Inverse Trigonometric Ratios ::逆三角比

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Resources ::资源

Inverse Trigonometric Ratios
::逆三角比

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)

Resources
::资源