Geometria e trigonometria
International Baccalaureate · IB Diploma · Mathematics: Analysis and Approaches · HL Tópico 3 4:06 Narração em inglês · Legendas em inglês + 中文 gravadas
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Transcrição
These are selected topics, not the whole syllabus, and every example is fictional.
这些是精选主题，不是完整大纲，所有例子都是虚构的。
Similar shapes scale lengths by k, areas by k squared and volumes by k cubed.
相似图形：长度乘 k，面积乘 k²，体积乘 k³。
Right triangles use Pythagoras and SOH CAH TOA; other triangles use the sine and cosine rules.
直角三角形用勾股定理和 SOH CAH TOA；其他三角形用正弦和余弦定理。
In radians an arc is r theta, and an equation has every solution in the interval.
弧度制下弧长是 rθ；方程要写出区间内的所有解。
A vector has magnitude and direction; AB is b minus a.
向量有大小和方向；AB = b − a。
A model has lengths one third of the real object, so the length scale factor to the real thing is 3.
模型的长度是实物的三分之一，所以到实物的长度比例因子是 3。
Area goes up by 3 squared, 9: a model area of 12 square centimetres means 108 on the real object.
面积乘 3² = 9：模型面积 12 平方厘米，实物是 108。
Volume goes up by 3 cubed, 27: 8 cubic centimetres becomes 216.
体积乘 3³ = 27：8 立方厘米变成 216。
A cylinder of radius 3 and height 5 is a prism with a circular cross-section.
半径 3、高 5 的圆柱是横截面为圆的柱体。
The base area is pi times 3 squared, 9 pi.
底面积是 π × 3² = 9π。
Stack it through the height 5: the volume is 45 pi cubic units, about 141.4.
沿高 5 叠起来：体积 45π 立方单位，约 141.4。
Keep pi until the last step.
π 留到最后一步。
A ladder reaches 4 metres up a wall with its foot 3 metres out.
梯子靠墙，顶端高 4 米，底端离墙 3 米。
Pythagoras gives its length: root of 9 plus 16, 5 metres.
勾股定理给出梯长：√(9 + 16) = 5 米。
For the angle with the ground, label from that angle: 4 is opposite, 3 is adjacent.
求与地面的夹角，从这个角标边：4 是对边，3 是邻边。
So tan theta is 4 over 3 and theta is about 53.1 degrees.
所以 tan θ = 4/3，θ 约 53.1°。
Check the calculator is in degrees.
检查计算器在角度制。
Two sides of 6 and 8 enclose 60 degrees, with no right angle anywhere, so Pythagoras does not apply.
两边 6 和 8 夹 60°，没有直角，所以不能用勾股定理。
The cosine rule corrects it: c squared is 36 plus 64 minus 2 times 6 times 8 times cos 60, which is 52.
余弦定理加以修正：c² = 36 + 64 − 2 × 6 × 8 × cos60° = 52。
So c is root 52, about 7.21.
所以 c = √52，约 7.21。
The area is a half times 6 times 8 times sin 60: 12 root 3.
面积是 ½ × 6 × 8 × sin60° = 12√3。
A wheel of radius 4 turns through pi over 3 radians.
半径 4 的轮子转过 π/3 弧度。
In radians the arc is simply r times theta: 4 pi over 3.
弧度制下弧长就是 r × θ：4π/3。
The sector area is a half r squared theta: a half times 16 times pi over 3, which is 8 pi over 3.
扇形面积是 ½r²θ：½ × 16 × π/3 = 8π/3。
With degrees you would need theta over 360 instead.
用角度制时要改用 θ/360。
Solve sine x equals a half for x from 0 to 2 pi.
在 0 到 2π 内解 sin x = ½。
The calculator gives pi over 6, the principal value.
计算器给出主值 π/6。
But the line y equals a half cuts the sine curve twice in a turn; by symmetry the other is pi minus pi over 6, which is 5 pi over 6.
但直线 y = ½ 在一周内与正弦曲线交两次；由对称性另一个解是 π − π/6 = 5π/6。
For sine 2 x, solve 2 x over 0 to 4 pi first, then halve.
解 sin2x 时，先在 0 到 4π 内解 2x，再除以 2。
Walk 4 metres east, then 3 metres north.
向东走 4 米，再向北走 3 米。
You have travelled 7 metres, a distance.
走过的路程是 7 米。
But the displacement, the straight arrow from start to finish, is 4, 3, of magnitude root of 16 plus 9: 5 metres.
但位移是从起点到终点的直箭头 (4, 3)，大小 √(16 + 9) = 5 米。
A vector has magnitude and direction; a distance has only size.
向量有大小和方向；路程只有大小。
Take a equals 4, 1 and b equals 1, 3.
取 a = (4, 1)，b = (1, 3)。
Put b's tail at a's tip: the sum is 5, 4, adding matching components.
把 b 的起点放在 a 的终点：和是 (5, 4)，对应分量相加。
The scalar product multiplies matching components and adds: 4 times 1 plus 1 times 3, which is 7.
数量积是对应分量相乘再相加：4 × 1 + 1 × 3 = 7。
And reversing a vector changes every sign: BA is minus AB.
把向量反向，每个分量都变号：BA = −AB。
Check yourself.
自我检测。
Doubling every length multiplies volume by 2 cubed: 8, not 2.
所有长度加倍，体积乘 2³ = 8，不是 2。
The cosine rule gives 36 plus 64 minus 48: c squared is 52.
余弦定理：36 + 64 − 48，c² = 52。
Sine x equals a half has two solutions in a turn: pi over 6 and 5 pi over 6.
sin x = ½ 在一周内有两个解：π/6 和 5π/6。
From A to B is 4, 3, of magnitude 5.
从 A 到 B 是 (4, 3)，大小为 5。