4.10 反向三角函数的应用
Section outline
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You are mapping out a hiking trip to a new spot in the state park near your home. On your map, you know that you are supposed to go on a course which ends with you having moved 2.5 miles East and 3 miles South. Your map, with the start and end points, is shown here:
::您正在绘制前往您家附近的州公园新地点的远足旅行图。 在您的地图上, 您知道您应该走上一条路线, 最终您已经向东移动了2.5英里,向南移动了3英里。 您的地图, 包括起点和终点, 请在此显示 :Now you need to calculate the angle that you need to walk with respect to due East. Can you find a way to calculate this angle using ?
::现在您需要计算您需要向东方向行走的角度。 您可以找到一个方法来计算这个角度 ?Applying Inverse Trig Functions
::应用逆三角函数The following problems are real-world problems that can be solved using the trigonometric functions . In everyday life, indirect measurement is used to obtain answers to problems that are impossible to solve using measurement tools. However, mathematics will come to the rescue in the form of trigonometry to calculate these unknown measurements.
::下列问题是使用三角函数可以解决的现实世界问题。 在日常生活中,间接测量被用来找到无法用测量工具解决问题的答案。然而,数学将以三角测量法的形式被拯救,以计算这些未知的测量数据。On a cold winter day the sun streams through your living room window and causes a warm, toasty atmosphere. This is due to the angle of inclination of the sun which directly affects the heating and the cooling of buildings. Noon is when the sun is at its maximum height in the sky and at this time, the angle is greater in the summer than in the winter. Because of this, buildings are constructed such that the overhang of the roof can act as an awning to shade the windows for cooling in the summer and yet allow the sun’s rays to provide heat in the winter. In addition to the construction of the building, the angle of inclination of the sun varies according to the latitude of the building’s location.
::在寒冷的冬天,太阳流穿过你客厅的窗户,造成温暖的烤热气氛。这是因为太阳的倾角直接影响到建筑物的供暖和冷却。 中午是太阳在天空最高高度的时候,这个角度在夏天比冬天大。 由于这个原因,建筑的屋顶上方可以遮盖夏天冷却的窗户,同时又让太阳的光线在冬天提供热。 除了建筑外,太阳的倾角也因建筑地点的纬度而不同。If the latitude of the location is known, then the following formula can be used to calculate the angle of inclination of the sun on any given date of the year:
::如果已知位置的纬度,则可以使用下列公式计算每年任何特定日期太阳倾角角:where represents the number of the day of the year that corresponds to the date of the year. Note: This formula is accurate to
::太阳的角=90latitude23.5cos[(N+10)360365],其中N代表与当年日期对应的当年天数。注意:这个公式精确到%12。Determine the measurement of the sun’s angle of inclination for a building located at a latitude of , March , the day of the year.
::确定位于年3月10日即年年第69天42°C纬度的建筑物的太阳倾角的测量。
::太阳的角=904223.5cos[(69+10)360.365] 太阳的角=4823.5(0.2093) 太阳的角=484.92太阳的角=43.08Determine the measurement of the sun’s angle of inclination for a building located at a latitude of , September .
::确定位于9月21日, 9月20日, 9月21日, 位于纬度的建筑物的太阳倾角的度量。
::太阳的角=90 20 23.5 cos[(264+10)360-365] 太阳的角=70 23.5(0.0043) 太阳的角=70.10Using inverse trigonometric functions
::使用逆三角函数1. A tower, 28.4 feet high, must be secured with a guy wire anchored 5 feet from the base of the tower. What angle will the guy wire make with the ground?
::1. 一座28.4英尺高的塔,必须用一个家伙的铁丝网固定在离塔底5英尺的地方。Draw a picture.
::画一张画。
::tanopp.adj.tan 28,45tan 5,68tan -1 (tan) =tan -1 (5.68) 80.02The following problem that involves functions and their inverses will be solved using the property . In addition, technology will also be used to complete the solution.
::以下涉及功能及其反向的问题将使用财产f(f-1(x))=f-1(f)(x)加以解决。 此外,还将使用技术来完成解决方案。2. In the main concourse of the local arena, there are several viewing screens that are available to watch so that you do not miss any of the action on the ice. The bottom of one screen is 3 feet above eye level and the screen itself is 7 feet high. The angle of vision (inclination) is formed by looking at both the bottom and top of the screen.
::2. 在地方舞台的主要汇合处,有若干屏幕可供观看,以免错过冰上的任何动作,一个屏幕的底部比眼睛水平高3英尺,屏幕本身高7英尺,视角(倾角)通过观察屏幕的底部和顶部形成。Sketch a picture to represent this problem.
::绘制一幅图来代表这个问题。Calculate the measure of the angle of vision that results from looking at the bottom and then the top of the screen. At what distance from the screen does the maximum value for the angle of vision occur?
::计算从观察屏幕底部到屏幕顶部得出的视觉角度的度量。 屏幕的距离是多少? 视觉角度的最大值是多少?
::2 = tan 1tan10x 和 tan1 = 3x2 = tan-1 (10x)- tan-1 (3x)To determine these values, use a graphing calculator and the trace function to determine when the actual maximum occurs.
::为确定这些值,使用图形计算器和跟踪函数来确定实际最大值何时发生。From the graph, it can be seen that the maximum occurs when . and .
::从图中可以看出,最大值发生在 x5.59英尺和32.57英寸时。Examples
::实例Example 1
::例1Earlier, you were asked to calculate the angle that you need to walk with respect to due East.
::早些时候,有人要求你计算需要步行到东端的角。You can set up a triangle that matches the physical situation of this problem. Here is what it should look like:
::您可以设置一个三角形, 该三角形与问题的物理环境相匹配 。 下面是它的外观 :Using the tangent function, you can solve for the angle you need to find:
::使用正切函数,您可以解析您需要找到的角 :
::-1(3-2.5) 50.19Example 2
::例2The intensity of a certain type of polarized light is given by the equation . Solve for .
::某类极分光的强度由I = I0sin =2 cos =222. 溶解 = 。
::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}我= {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}2... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}2... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080} {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080} {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}Example 3
::例3The following diagram represents the ends of a water trough. The ends are actually isosceles trapezoids, and the length of the trough from end-to-end is ten feet. Determine the maximum volume of the trough and the value of that maximizes that volume.
::下图代表水槽的尽端。 末端实际上为等骨质类, 端到端的槽长度为 10 英尺。 确定槽的最大体积和最大体积的 ° 值 。The volume is 10 feet times the area of the end. The end consists of two congruent right triangles and one rectangle. The area of each right triangle is and that of the rectangle is . This means that the volume can be determined by the function , and this function can be graphed as follows to find the maximum volume and the angle where it occurs.
::音量是末端面积的10英尺。 末端由两个相匹配的右三角形和一个矩形组成。 每个右三角形的面积是 12 (sin) (cos) , 矩形的面积是 (1)(cos) 。 这意味着音量可以由函数 V () = 10 (cossinçççç) 确定, 此函数可以如下图绘制, 以查找最大音量和角 。Therefore, the maximum volume is approximately 13 cubic feet and occurs when is about .
::因此,最大体积约为13立方英尺,当 约30立方英尺时发生。Example 4
::例4A boat is docked at the end of a 10 foot pier. The boat leaves the pier and drops anchor 230 feet away 3 feet straight out from shore (which is perpendicular to the pier). What was the bearing of the boat from a line drawn from the end of the pier through the foot of the pier?
::一艘船停靠在10英尺码头的尽头,船离开码头,在离岸直达3英尺(直通码头)的230英尺远的230英尺处投下锚。 从码头尽头到码头脚的一条线上,船只的轴承是什么?
::COSx=7230x=cos-17230x=88.26Review
::回顾-
The distance from a boat to a lighthouse is 100 feet and the lighthouse is 120 feet tall. What is the angle of depression from the top of the Lighthouse to the boat.
::从船到灯塔的距离是100英尺,灯塔是120英尺高,从灯塔的顶部到船的压抑角度是什么? -
You are standing 100 feet from an arch that is 68 feet tall. At what angle do you have to look up to see the top of the arch? Assume you are 5 feet tall.
::你距离一个68英尺高的拱门站着100英尺。你必须从什么角度向上看拱门的顶部?假设你5英尺高。 -
The angle of elevation of the top of a church to a point 100 feet away from the base is
. Find the height of the church.
::教堂顶部高到距离基地100英尺的高度角是60英尺。 找到教堂的高度。
You are standing looking at a large painting on the wall. The bottom of the painting is 1 foot above your eye level. The painting is 10 feet tall. Assume you are standing x feet from the painting and that angle is formed by the lines of vision to the bottom and to the top of the painting.
::您正站在墙上看着一幅大画。 画底部在您眼睛的高度上方一英尺。 画面是10英尺高的。 假设您站在画面上方的 x 英尺, 角由图画底部和顶部的视觉线形成 。-
Draw a picture to represent this situation.
::绘制图片来代表这种情况 。 -
Solve for
in terms of x.
::以x为单位 解决 . . . . . . . . -
If you are standing 10 feet from the painting, what is
?
::如果你站在离画十尺远的地方, 什么是? -
If
, how far are you standing from the wall (to the nearest foot)?
::如果"30",你站在墙上多远?
You are watching a hot-air balloon that was 300 feet from you when it started rising from the ground. Assume the height of the balloon is x and is the angle of elevation from the ground where you are standing up to the balloon.
::您正在观看一个热空气气球, 当气球开始从地面升起时距离您300英尺。 假设气球的高度是 x 和 ° 是一个从地面向气球站立的高度角度。-
Solve for x in terms of
.
::解决 x 的x 值 。 -
Solve for
in terms of x.
::以x为单位 解决 . . . . . . . . -
What is the angle of elevation when the hot-air balloon is 500 feet above the ground?
::当热空气气球在地上500英尺处时,高空的角度是什么? -
How high above the ground is the balloon when the angle of elevation is
?
::升降角是80°Q时 气球在地面上有多高?
Recall that if the latitude of the location is known, then the following formula can be used to calculate the angle of inclination of the sun on any given date of the year:
::回顾如果知道位置的纬度,可使用下列公式计算每年任何特定日期太阳倾角角:where represents the number of the day of the year that corresponds to the date of the year.
::太阳角=90latitude23.5cos[(N(N+10)360-365],其中N代表与年日期对应的当年天数。-
Determine the measurement of the sun’s angle of inclination for a building located at a latitude of
, April
, the
day of the year.
::确定位于年4月12日即年年第102天纬度30°Q的建筑物的太阳倾角的测量。 -
Determine the measurement of the sun’s angle of inclination for a building located at a latitude of
, August
, the
day of the year.
::确定位于8月14日,即每年第226天的纬度50°Z的建筑物的太阳倾角。
-
A tower, 50 feet high, is secured with a guy wire anchored 8 feet from the base of the tower. What angle will the guy wire make with the ground?
::一座50英尺高的塔 被一个铁丝钉固定在离塔底8英尺的地方 -
A 30 foot tall flagpole casts a 12 foot shadow. What is the angle that the sun hits the flagpole?
::一只30英尺高的旗杆投下12英尺的阴影。太阳击中旗杆的角度是什么?
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 。 -
The distance from a boat to a lighthouse is 100 feet and the lighthouse is 120 feet tall. What is the angle of depression from the top of the Lighthouse to the boat.