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피어슨 에덱셀 · International GCSE · Mathematics A 주제 4 5:44 영어 내레이션 · 영어 + 중국어 자막 burned-in
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Topic 4 is the largest topic.
第 4 主题是最大的主题。
The Higher formulae sheet gives the sine and cosine rules, the area half a b sine C, and the cone and sphere formulae.
高级卷公式表给出正弦定理、余弦定理、面积 ½ab sin C 以及圆锥和球的公式。
Arcs, sectors and the similarity ratios k, k squared and k cubed you recall yourself.
弧长、扇形和相似比 k、k²、k³ 要自己记住。
Take a regular pentagon and extend a side.
取一个正五边形，延长一条边。
The five exterior angles make one full turn, 360 degrees, so each is 72.
五个外角合起来转一整圈 360°，所以每个是 72°。
The interior angle is the rest of the straight line: 108.
内角是平角剩下的部分：108°。
As a check, the interior sum is 5 minus 2 times 180, which is 540, and 5 times 108 is 540.
检验：内角和是 (5 − 2) × 180 = 540，而 5 × 108 = 540。
Bearings are measured clockwise from north and written with three figures: 040, not 40.
方位角从北方顺时针量，写成三位数：040，而不是 40。
Q bears 040 from P.
Q 在 P 的 040° 方向。
Stand at Q with its own north line and look back: P bears 220, because the return bearing adds 180.
站在 Q 点，画出它自己的指北线，回头看：P 在 220° 方向，因为反方位角要加 180°。
Compound measures work the same careful way: 234 kilometres in 2.6 hours is 90 kilometres per hour.
复合量同样要细心：2.6 小时行 234 千米，就是每小时 90 千米。
Points 5 from O form a circle.
到 O 距离为 5 的点组成一个圆。
Points equidistant from 0, 0 and 6, 0 form the perpendicular bisector, x equals 3.
到 (0, 0) 和 (6, 0) 距离相等的点组成垂直平分线 x = 3。
Only where both hold do they meet: 3, 4 and 3, minus 4.
两个条件都满足的只有交点：(3, 4) 和 (3, −4)。
With straight edge and compasses, leave the construction arcs in.
用直尺和圆规作图时，要保留作图弧线。
A, B, C and D lie on a circle, and BCD is 128 degrees.
A、B、C、D 在圆上，角 BCD = 128°。
Opposite angles of a cyclic quadrilateral add to 180, so BAD is 52.
圆内接四边形的对角互补，所以 BAD = 52°。
The angle at the centre is twice the angle at the circumference, so BOD is 104.
圆心角是圆周角的两倍，所以 BOD = 104°。
OB and OD are radii, so triangle OBD is isosceles, and OBD is 38.
OB 和 OD 是半径，所以三角形 OBD 是等腰三角形，OBD = 38°。
Angle A is 40, angle B is 65, and side a is 8.
角 A = 40°，角 B = 65°，边 a = 8。
We know a side with its opposite angle, so use the sine rule: b over sine 65 equals 8 over sine 40.
已知一条边和它的对角，所以用正弦定理：b/sin 65° = 8/sin 40°。
Multiply up: b is 11.3 centimetres.
两边相乘：b = 11.3 厘米。
Look at the sine curve and the cosine curve from 0 to 180 degrees.
看 0° 到 180° 的正弦曲线和余弦曲线。
The sine curve is symmetric about 90, so sine stays positive: sine 130 equals sine 50.
正弦曲线关于 90° 对称，所以正弦保持为正：sin 130° = sin 50°。
The cosine curve falls through zero, so cosine turns negative: cosine 130 is minus cosine 50.
余弦曲线穿过零，所以余弦变成负数：cos 130° = −cos 50°。
That sign decides the cosine rule for an obtuse angle.
这个符号决定了钝角时的余弦定理。
Sides 7 and 9 meet at 110 degrees.
边 7 和 9 夹角 110°。
The cosine rule gives a squared equals 49 plus 81 minus 126 cosine 110.
余弦定理：a² = 49 + 81 − 126 cos 110°。
Cosine 110 is negative, so the last term is added: 173.1.
cos 110° 是负数，所以最后一项变成加：173.1。
So a is 13.2 centimetres, the longest side, facing the obtuse angle.
所以 a = 13.2 厘米，是最长的边，正对钝角。
A cuboid 6 by 8 by 10.
一个 6 × 8 × 10 的长方体。
The base diagonal is the first right triangle: root of 6 squared plus 8 squared, which is 10.
底面对角线是第一个直角三角形：√(6² + 8²) = 10。
The space diagonal stands on it with the height 10: root 200, about 14.1.
体对角线以它为底，高为 10：√200，约 14.1。
The angle with the base has tan theta equal to 10 over 10, so theta is 45 degrees.
它与底面的夹角满足 tan θ = 10/10，所以 θ = 45°。
In the box, each of BM, BP and MP comes from its own Pythagoras: root 200, root 244 and 10.
在长方体中，BM、BP、MP 各由一次勾股定理得到：√200、√244 和 10。
Now triangle BMP is an ordinary triangle with three known sides.
现在三角形 BMP 就是三边已知的普通三角形。
The cosine rule gives cosine BMP equal to 0.198, so angle BMP is 78.6 degrees.
余弦定理得 cos BMP = 0.198，所以角 BMP = 78.6°。
The hexagon splits along FC into a rectangle and a trapezium.
沿 FC 把六边形分成一个长方形和一个梯形。
The rectangle is 23 by 4.7: 108.1.
长方形 23 × 4.7 = 108.1。
The trapezium's height comes from the slant side: 5 sine 30, which is 2.5, so its area is a half of 11 plus 23, times 2.5: 42.5.
梯形的高来自斜边：5 sin 30° = 2.5，所以面积是 ½(11 + 23) × 2.5 = 42.5。
Together, 150.6 square centimetres.
合计 150.6 平方厘米。
Slice the cone through its axis.
沿轴切开圆锥。
Radius 3 and height 4 make a right triangle, so the slant is 5.
半径 3 和高 4 构成直角三角形，所以斜高是 5。
The volume is one third pi r squared h: 12 pi.
体积是 ⅓πr²h：12π。
The curved surface is pi r l: 15 pi.
侧面积是 πrl：15π。
A sector with radius 9 and 40 degrees has arc 2 pi and area 9 pi.
半径 9、圆心角 40° 的扇形，弧长 2π，面积 9π。
Two similar bottles have surface areas 180 and 405.
两个相似的瓶子表面积分别为 180 和 405。
The area ratio is 2.25, which is k squared.
面积比 2.25 就是 k²。
So the length scale factor is 1.5.
所以长度比是 1.5。
Now cube it: 3.375.
再立方：3.375。
The larger bottle holds 320 times 3.375, which is 1080 cubic centimetres.
大瓶容积是 320 × 3.375 = 1080 立方厘米。
Multiplying by 2.25 would give the common wrong answer, 720.
乘以 2.25 会得到常见的错误答案 720。
Check yourself.
自我检测。
A regular 10-gon has exterior angle 36 degrees.
正十边形的外角是 36°。
The return bearing of 040 is 220.
040° 的反方位角是 220°。
Opposite a 128 degree angle in a cyclic quadrilateral is 52.
圆内接四边形中 128° 的对角是 52°。
Cosine 110 is negative.
cos 110° 是负数。
And an area ratio of 4 to 9 means a length ratio of 2 to 3.
面积比 4:9 意味着长度比 2:3。