Trigonometry
IGCSE Mathematics Topic 6 8:37 English narration · English + 中文 subtitles burned in
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Transcript
How tall is that tower?
这座塔有多高？
You cannot climb it with a tape measure.
你没法带着卷尺爬上去。
But you can stand on the ground, walk fifty metres back, and measure one angle — the angle you look up through.
但你可以站在地面上，往后走五十米， 量出一个角——你抬头看过去的那个角。
That one angle is enough.
这一个角就足够了。
By the end of this lesson you will find the height in a single line of working.
学完这节课，你只用一行算式就能求出它的高度。
Trigonometry links the angles in a triangle to the lengths of its sides.
三角学把三角形的角和边长联系起来。
First Pythagoras, for sides only.
先是勾股定理，只涉及边。
Then the three ratios for right-angled triangles.
然后是直角三角形的三个比值。
Then elevation and depression, the exact values, the wave graphs, and two rules for any triangle.
接着是仰角和俯角、精确值、波形图像， 以及适用于任意三角形的两条定理。
Parts marked Extended are for the Extended papers only.
标注 Extended 的部分只在扩展卷中考查。
Pythagoras' theorem works in one special triangle: the right-angled triangle.
勾股定理（Pythagoras' theorem）只适用于一种特殊的三角形：直角三角形。
Its longest side is the hypotenuse, always opposite the right angle.
它最长的边是斜边， 总在直角的对面。
Watch why it is true.
来看它为什么成立。
One square, four identical triangles.
一个正方形，四个全等三角形。
Slide them across and the space left over cannot change — but now it is two squares, not one.
把它们滑动一下，剩下的空白面积不会变——但现在它是两个正方形，不是一个。
So the square on the hypotenuse equals the other two.
所以斜边上的正方形，等于另外两个。
First exam question.
第一道考题。
A right-angled triangle has a hypotenuse of thirteen centimetres and one short side of five centimetres. Find the other short side.
一个直角三角形，斜边十三厘米，一条短边五厘米，求另一条短边。
Pause the video and work it out.
先暂停视频，自己算一算。
Which side is missing?
缺的是哪一条边？
We already know the longest side, so this time we subtract.
我们已经知道最长的边， 所以这次要用减法。
Thirteen squared is one hundred and sixty-nine. Five squared is twenty-five.
十三的平方是一百六十九，五的平方是二十五。
Take them away and you get one hundred and forty-four.
相减得到一百四十四。
The square root of that is twelve centimetres.
它的平方根是十二厘米。
Pythagoras needs two known sides.
勾股定理需要已知两条边。
But most questions give you an angle instead.
但大多数题目给的是一个角。
Then you need three new ratios — and first you must name the sides.
这时你需要三个新的比值——而首先要给各边起名字。
Always name them from the angle you are using.
永远从你正在用的那个角出发命名。
The side across from that angle is the opposite.
与这个角相对的边是对边。
The side from the angle to the right angle is the adjacent.
从这个角连到直角的那条边是邻边。
Change the angle, and those two swap over.
换一个角，这两条边就互换。
Now the ratios.
现在看这三个比值。
Sine is the opposite divided by the hypotenuse.
正弦等于对边除以斜边。
Cosine is the adjacent divided by the hypotenuse.
余弦等于邻边除以斜边。 正切等于对边除以邻边。
Tangent is the opposite divided by the adjacent. Remember them as soh cah toa.
用 SOH CAH TOA 来记。
Watch the triangle grow and shrink. Every length changes, but the ratio does not move.
看三角形放大又缩小： 每条边都在变，但比值不动。
The ratio belongs to the angle, not to the size.
比值属于这个角，与大小无关。
Two quick examples.
两个小例子。
First finding a side.
第一，求边长。
The hypotenuse is ten centimetres and the angle is thirty degrees.
斜边十厘米，角是三十度。
We want the opposite side, so we need opposite over hypotenuse — that is sine.
我们要求对边， 所以用对边比斜边——也就是正弦。
Ten times the sine of thirty degrees gives five centimetres.
十乘以三十度的正弦，得到五厘米。
Second finding an angle.
第二，求角度。
The opposite is four, the adjacent is three, so we use tangent.
对边是四，邻边是三，所以用正切。
To turn a ratio back into an angle, press the inverse tangent key.
要把比值变回角度，就按反正切键。
That gives fifty-three point one degrees.
结果是五十三点一度。
Real questions rarely say the word sine. They talk about looking up and looking down.
真实的题目很少说正弦这个词，它们说的是抬头看和低头看。
The angle of elevation is measured up from the horizontal.
仰角是从水平方向向上量的角。
The angle of depression is measured down from the horizontal — never from the vertical.
俯角是从水平方向向下量的角——绝不是从竖直方向量。
So draw the horizontal line first, every time.
所以每次都要先画那条水平线。
And remember: the shortest distance from a point to a line is the perpendicular distance.
还要记住：从一点到一条直线的最短距离是垂直距离。
Now back to our tower.
现在回到我们的塔。
You stand fifty metres from its foot, and the angle of elevation to the top is forty degrees.
你站在离塔脚五十米处，看塔顶的仰角是四十度。
Sketch it first.
先画草图。
The fifty metres lies along the ground next to the angle, so it is the adjacent.
五十米沿着地面，紧挨着这个角，所以它是邻边。
The height stands across from the angle, so it is the opposite.
高度在这个角的对面， 所以它是对边。
Opposite over adjacent is tangent.
对边比邻边就是正切。
That tangent is zero point eight three nine, and fifty times it is forty-two metres.
这个正切是零点八三九， 五十乘以它，等于四十二米。
Some angles you must know without a calculator.
有些角度必须不用计算器就知道。
They come from two small triangles.
它们来自两个小三角形。
Cut a square of side one along its diagonal: you get two forty-five degree angles, and a diagonal of the square root of two.
把边长为一的正方形沿对角线剪开：得到两个四十五度角，对角线是根号二。
Cut an equilateral triangle of side two in half: thirty and sixty degrees, with a height of the square root of three.
把边长为二的等边三角形从中间剪开：得到三十度和六十度，高是根号三。
Where do the trig graphs come from?
三角函数的图像从哪里来？
Imagine a seat on a Ferris wheel turning at a steady speed.
想象摩天轮上的一个座位，以匀速转动。
Watch its height above the centre as the wheel goes round.
看它相对圆心的高度如何随转动变化。
Plot that height against the angle turned, and a smooth wave appears.
把高度对转过的角度画出来， 就出现一条平滑的波。
It rises to one at ninety degrees, drops to minus one at two hundred and seventy, and returns to zero at three hundred and sixty.
它在九十度升到一，在二百七十度降到负一， 在三百六十度回到零。
That wave is the sine graph.
这条波就是正弦图像。
Here are both waves together, from zero to three hundred and sixty degrees.
这里是两条波放在一起，从零度到三百六十度。
Sine starts at zero.
正弦从零开始。
Cosine has the same shape but starts at one — it is sine shifted left by ninety degrees.
余弦形状相同，但从一开始——它就是正弦向左平移九十度。
Both stay between minus one and one.
两者都在负一和一之间。
The tangent graph is different: it climbs steeply, breaks, and repeats every one hundred and eighty degrees.
正切图像不同：它陡峭上升，然后中断， 每一百八十度重复一次。
Now solve a trigonometric equation.
现在解一个三角方程（trigonometric equation）。
Find every angle between zero and three hundred and sixty degrees whose sine is the square root of three, over two.
求零度到三百六十度之间，正弦值等于二分之根号三的所有角。
Your calculator gives one answer: sixty degrees.
计算器只给一个答案：六十度。
But look at the graph — the flat line cuts the wave twice.
但看这张图——那条水平线和波交于两点。
The second crossing is the mirror image, at one hundred and eighty take away sixty, which is one hundred and twenty.
第二个交点是它的镜像，在一百八十减六十，也就是一百二十度。
So there are two answers.
所以有两个答案。
Now a cosine equation, because the mirror rule changes.
现在来看一个余弦方程，因为镜像规则变了。
Solve two cosine x, plus one, equals zero.
解 二 cos x 加一 等于零。
It is not in the right shape yet, so rearrange it first: cosine x equals minus a half.
它还不是需要的形式， 所以先重新整理：cos x 等于负二分之一。
Now the calculator gives one hundred and twenty degrees.
现在计算器给出一百二十度。
Here is where students lose the second mark.
学生就是在这里丢掉第二 个分。
With sine we mirrored in one hundred and eighty minus x.
对正弦，我们用一百八十减 x 做镜像。
Do that here and you get sixty, which is wrong — the cosine of sixty is plus a half, not minus a half.
在这里这样做会得到六十，那是错的——六十度的 余弦是正二分之一，不是负二分之一。
Look at the graph instead: the cosine curve is symmetric about the vertical axis, so its second crossing sits at three hundred and sixty, minus the first one.
看图像：余弦曲线关于纵轴对称，所以它的第二个交点在 三百六十减去第一个解的位置。
Three hundred and sixty take away one hundred and twenty is two hundred and forty.
三百六十减一百二十等于二百四十。
So the answers are one hundred and twenty and two hundred and forty.
所以答案是一百二十度和 二百四十度。
Same method, a different rule for the second solution: sine mirrors in one eighty minus x, cosine mirrors in three sixty minus x.
方法相同，但第二个解的规则不同：正弦用一百八十减 x，余弦用三百六十减 x。
So far every triangle had a right angle. Most do not.
到目前为止每个三角形都有直角，但大多数没有。
For any triangle, label the sides with small letters and the opposite angles with matching capitals.
对任意三角形，用小写字母标边，用相对的大写字母标角。 然后两条定理就够了。
Then two rules cover everything. Use the sine rule when you have an angle with the side opposite it — and watch for the ambiguous case, where that angle could be acute or obtuse.
当你有一个角和它的对边时，用正弦定理——并留意两解情况（ambiguous case），那个角可能是锐角也可能是钝角。
Use the cosine rule when you have two sides and the angle between them.
当你有两条边和它们的夹角时，用余弦定理。
A cosine rule question.
一道余弦定理的题。
Two sides are seven and eight centimetres, and the angle between them is forty degrees. Find the third side.
两条边是七厘米和八厘米，夹角四十度，求第三条边。
Square the two sides and add: forty-nine plus sixty-four is one hundred and thirteen.
把两条边平方相加：四十九加六十四等于一百一十三。
Now subtract twice their product times the cosine of forty degrees. That leaves twenty-seven point two.
再减去它们乘积的两倍乘以四十度的余弦。
Take the square root, and the third side is five point two centimetres.
剩下二十七点二。 开平方，第三条边就是五点二厘米。
From the same numbers the area is eighteen square centimetres.
用同样这几个数，面积是十八平方厘米。
Last piece: three dimensions.
最后一部分：三维。
A common task is the angle between a line and a plane.
常见题是求一条直线与一个平面的夹角。
The trick never changes — find a flat right-angled triangle inside the solid.
方法始终不变——在立体图形里找出一个平面直角三角形。
Take a box six by eight on the base, five tall, and find the angle between a space diagonal and the base.
取一个底面六乘八、高五的长方体，求空间对角线与底面的夹角。
Start with the base diagonal: six squared plus eight squared is one hundred, so it is ten.
先求底面对角线：六的平方加八的平方等于一百，所以是十。
That diagonal and the height make a right-angled triangle, so the tangent is five over ten, and the angle is twenty-six point six degrees.
这条对角线和高构成一个直角三角形，正切是五比十，角就是二十六点六度。
Three habits that save marks.
三个保分习惯。
First, no angle in the question means Pythagoras; an angle means soh cah toa.
第一，题目里没有角就用勾股定理；有角就用 SOH CAH TOA。
Second, label the opposite, the adjacent and the hypotenuse on your sketch before you pick a ratio.
第二，选比值之前，先在草图上标出对边、邻边和斜边。
Third, check that your calculator is in degrees.
第三，检查计算器是不是角度模式。
Get these right, and trigonometry is yours.
做好这三点，三角学就是你的了。