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국제바칼로레이트 · IB Diploma · Mathematics: Analysis and Approaches · SL 주제 3 3:12 영어 내레이션 · 영어 + 중국어 자막 burned-in
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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 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。
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。