Course: 4.2 反弦、余弦和唐恩特的精确值

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反正弦、余弦和唐光的精确值
You are working with a triangular brace in shop class. The brace is a right triangle, and the length of one side of the bracket is $\begin{aligned} \sqrt{3} \approx 1.732 \end{aligned}$ and it is connected to the other side at a right angle. The length of the other side is $\begin{aligned} 1 \end{aligned}$ . You need to find the angle that the third piece makes with the first piece, labelled below as "C":
::您在商店类中正在使用三角括号。括号是一个右三角形, 括号的一方长度为 3 1.732, 并用右角度连接到另一侧。另一侧的长度为 1 。您需要找到第三段与第一个段( 以下标为“ C ” ) 的角 :

Can you find the angle between the legs of the brace?
::你能在支架的两腿之间找到角度吗?

By the time you finish reading this lesson, you'll be able to answer this question.
::当你读完这堂课你就能回答这个问题了

Inverse of Sine, Cosine and Tangent
::松树、脊弦和唐恩特的反面

can be useful in a variety of math problems for finding angles that you need to know. In many cases, such as angles involving multiples of $\begin{aligned} 30^{\circ}, 60^{\circ} \end{aligned}$ and $\begin{aligned} 90^{\circ} \end{aligned}$ , the values of trig functions are often memorized, since they are used so often.
::在各种数学问题中,找到您需要知道的角度是有用的。在很多情况下,比如30+,60++和90++的多重角度,三角函数的值往往被记住,因为它们经常被使用。

Recall the unit circle and the critical values. With the inverse trigonometric functions , you can find the angle value (in either radians or degrees) when given the ratio and function. Make sure that you find all solutions within the given interval.
::回想单位圆和关键值。如果使用逆三角函数,在给定比例和函数时,您可以找到角值(以弧度或度计)。确保您在给定间隔内找到所有解决方案。

Let's take a look at a few example problems.
::让我们来看一下几个例子问题。

1. Find the exact value of the expression without a calculator, in $\begin{aligned} [0, 2 π) \end{aligned}$ .
::1. 在[0,2]中查找无计算器表达式的准确值。

$\begin{aligned} \sin^{- 1} (- \frac{\sqrt{3}}{2}) \end{aligned}$
:-32) (-32)

This is a value from the special right triangles and the unit circle.
::这是特殊右三角形和单位圆的数值。

Recall that $\begin{aligned} - \frac{\sqrt{3}}{2} \end{aligned}$ is from the $\begin{aligned} 30 - 60 - 90 \end{aligned}$ triangle. The reference angle for $\begin{aligned} \sin \end{aligned}$ and $\begin{aligned} \frac{\sqrt{3}}{2} \end{aligned}$ would be $\begin{aligned} 60^{\circ} \end{aligned}$ . Because this is sine and it is negative, it must be in the third or fourth quadrant. The answer is either $\begin{aligned} \frac{4 π}{3} \end{aligned}$ or $\begin{aligned} \frac{5 π}{3} \end{aligned}$ .
::回顾- 32 是来自 30- 60- 90 三角形。罪与 32 的参考角度是 60 。由于这是正弦和负的, 它必须在 3 或 4 4 中。答案是 4 3 或 5 3 。

2. Find the exact value of the expression without a calculator, in $\begin{aligned} [0, 2 π) \end{aligned}$ .
::2. 在[0,2]中查找无计算器表达式的准确值。

$\begin{aligned} \cos^{- 1} (- \frac{\sqrt{2}}{2}) \end{aligned}$
::COS-1(-22)

This is a value from the special right triangles and the unit circle.
::这是特殊右三角形和单位圆的数值。

$\begin{aligned} - \frac{\sqrt{2}}{2} \end{aligned}$ is from an isosceles right triangle. The reference angle is then $\begin{aligned} 45^{\circ} \end{aligned}$ . Because this is cosine and negative, the angle must be in either the second or third quadrant. The answer is either $\begin{aligned} \frac{3 π}{4} \end{aligned}$ or $\begin{aligned} \frac{5 π}{4} \end{aligned}$ .
::~ 22 是来自一个等星向右三角形。引用角度为 45\\\\\\\\\\\ x\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\可以\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

3. Find the exact value of the expression without a calculator, in $\begin{aligned} [0, 2 π) \end{aligned}$ .
::3. 在[0,2]中查找无计算器表达式的确切值。

$\begin{aligned} \tan^{- 1} \sqrt{3} \end{aligned}$
::坦 - 1 +% 3

This is a value from the special right triangles and the unit circle.
::这是特殊右三角形和单位圆的数值。

$\begin{aligned} \sqrt{3} \end{aligned}$ is also from a $\begin{aligned} 30 - 60 - 90 \end{aligned}$ triangle. Tangent is $\begin{aligned} \sqrt{3} \end{aligned}$ for the reference angle $\begin{aligned} 60^{\circ} \end{aligned}$ . Tangent is positive in the first and third quadrants, so the answer would be $\begin{aligned} \frac{π}{3} \end{aligned}$ or $\begin{aligned} \frac{4 π}{3} \end{aligned}$ .
::3 也来自 30- 60- 90 三角形。引用角度 60 的唐恩特为 3。第一和第二夸的唐恩特是正数, 所以答案是 3 或 4 3 。

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the angle between the legs of the brace.
::早些时候,有人要求你找到支架两腿之间的角

Using your knowledge of the values of trig functions for angles, you can work backward to find the angle that the brace makes:
::使用您对角度三角函数值的知识,您可以向后工作,找到括号所设定的角:

$\begin{aligned} \tan C = \frac{1}{\sqrt{3}} \\ \tan^{- 1} C = \tan^{- 1} \frac{1}{\sqrt{3}} \\ C = 60^{\circ} \end{aligned}$

::坦 - 1 - 1 - 1 - 1 - 1 - 13 - 13 C= 60

Example 2
::例2

Find the exact value of the inverse function of $\begin{aligned} \cos^{- 1} (0) \end{aligned}$ , without a calculator in $\begin{aligned} [0, 2 π) \end{aligned}$
::在 [0,2] 没有计算器的情况下, 查找 cos- 1\\\( 0) 的反函数的准确值。

$\begin{aligned} \frac{π}{2}, \frac{3 π}{2} \end{aligned}$

Example 3
::例3

Find the exact value of the inverse function of $\begin{aligned} \tan^{- 1} (- \sqrt{3}) \end{aligned}$ , without a calculator in $\begin{aligned} [0, 2 π) \end{aligned}$
::在 [0,2] 中查找tan-1(- 3) 的反函数的准确值,但不使用计算器

$\begin{aligned} \frac{2 π}{3}, \frac{5 π}{3} \end{aligned}$

Example 4
::例4

Find the exact value of the inverse function of $\begin{aligned} \sin^{- 1} (- \frac{1}{2}) \end{aligned}$ , without a calculator in $\begin{aligned} [0, 2 π) \end{aligned}$
::查找 sin -1 (- 12) 的反函数的准确值, 在 [ 0, 2) 没有计算器

$\begin{aligned} \frac{11 π}{6}, \frac{7 π}{6} \end{aligned}$

Review
::回顾

Find the exact value of each expression without a calculator, in $\begin{aligned} [0, 2 π) \end{aligned}$ .
::在 [0,2] 中查找没有计算器的每个表达式的确切值。

$\begin{aligned} \sin^{- 1} (\frac{\sqrt{2}}{2}) \end{aligned}$
:22) (22)

$\begin{aligned} \cos^{- 1} (\frac{1}{2}) \end{aligned}$
::COS-1( 12)

$\begin{aligned} \sin^{- 1} (1) \end{aligned}$
::第1条第1款(1)

$\begin{aligned} \cos^{- 1} (- \frac{\sqrt{3}}{2}) \end{aligned}$
:-32) (c) (-) (-) (-) (-) (-) (-) (-) (-) (-) (-) (-) (- (-) (-) (-) (-) (-) (-) (-)

$\begin{aligned} \tan^{- 1} (- \frac{\sqrt{3}}{3}) \end{aligned}$
::tan-1(-33)

$\begin{aligned} \tan^{- 1} (- 1) \end{aligned}$
::tan- 1 (- 1)

$\begin{aligned} \sin^{- 1} (\frac{\sqrt{3}}{2}) \end{aligned}$
:32)

$\begin{aligned} \cos^{- 1} (\frac{\sqrt{2}}{2}) \end{aligned}$
::COS-1(22)

$\begin{aligned} \csc^{- 1} (\sqrt{2}) \end{aligned}$
::csc-1_________________(2)

$\begin{aligned} \sec^{- 1} (- 2) \end{aligned}$
::秒- 1(- 2)

$\begin{aligned} \cot^{- 1} (\frac{\sqrt{3}}{3}) \end{aligned}$
:33)

$\begin{aligned} \sec^{- 1} (\frac{2 \sqrt{3}}{2}) \end{aligned}$
::秒-1(232)

$\begin{aligned} \csc^{- 1} (- \frac{2 \sqrt{3}}{2}) \end{aligned}$
::csc-1(-232)

$\begin{aligned} \cot^{- 1} (- \sqrt{3}) \end{aligned}$
:- 3) (- 3)

$\begin{aligned} \cot^{- 1} (- 1) \end{aligned}$
::COt- 1 (- 1)

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。

Section outline

General

反正弦、 余弦和唐光的精确值

Inverse of Sine, Cosine and Tangent ::松树、脊弦和唐恩特的反面

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

反正弦、余弦和唐光的精确值

Inverse of Sine, Cosine and Tangent
::松树、脊弦和唐恩特的反面

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)