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3 · 几何与三角学

国际文凭组织 · IB Diploma · 数学：分析与方法 · SL · 知识点 3

训练
3.1

Scope and prerequisites

Supported SL focus. First assessment 2021; current through 2028. First-assessment-2029 course is separate.. Remaining guide, assessment and practical requirements retain their recorded holds.

Prerequisites: read the stated quantities and units, use arithmetic and the model conditions below. Each lesson develops its own method before independent transfer.

These are original or explicitly fictional teaching examples, not actual measurements or completed assessed learner investigations.

3.2

角度推理、相似性与度量

Does volume scale like length?

  • A model has lengths one third of the real object. How much smaller are its area and volume?
  • This lesson studies scale factor 相似比: The multiplier that relates corresponding lengths in similar shapes.

Choose the mathematical structure

  • For similar shapes with length scale factor k, areas scale by k² and volumes by k³. State angle reasons explicitly. A circle's tangent is perpendicular to the radius at the contact point.
  • State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

$$\frac{A_2}{A_1}=k^2,\qquad \frac{V_2}{V_1}=k^3$$
  • Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Example:

If model-to-real length factor is 3, a model area of 12 cm² gives 12×3²=108 cm² and a model volume of 8 cm³ gives 8×3³=216 cm³. A cylinder with r=3,h=5 has volume πr²h=45π.

Angle reasoning, similarity and mensuration — original teaching diagram

Test a tempting shortcut

  • Equal angles alone establish similarity, not equal size. Use corresponding lengths in the same order. Convert linear units before calculating area or volume, or square/cube the conversion factor correctly.
  • When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Warn:

Doubling every length of a solid doubles its volume. This claim is false. Explain which definition or assumption it violates.

Interpret a new situation

  • A geometric proof should name the relevant theorem, identify the equal angle or ratio, and draw the conclusion. A scale drawing is evidence only when the task permits measurement.
  • A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

  • Current first-assessment-2021 Analysis and Approaches SL. This is authored concept support; the full guide is needed to certify every objective.
  • Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

Key:

The multiplier that relates corresponding lengths in similar shapes. Choose the relationship, show the method, check its assumptions and interpret the result.

词汇 训练
English 中文 拼音
scale factor/skeɪl ˈfæktə/ 相似比 xiāng sì bǐ
3.3

直角三角形与非直角三角形

Which side does the ladder need?

  • A ladder reaches a height of 4 m while its foot is 3 m from a wall. Which sides are known, and which angle do we need?
  • This lesson studies hypotenuse 斜边: The side opposite the right angle in a right-angled triangle.

Choose the mathematical structure

  • In a right triangle a²+b²=c²; sinθ=opposite/hypotenuse, cosθ=adjacent/hypotenuse and tanθ=opposite/adjacent. For other triangles, use the sine or cosine rule, or area=ab sin C/2.
  • State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

$$a^2+b^2=c^2,\qquad \tan\theta=\frac{\mathrm{opposite}}{\mathrm{adjacent}}$$
  • Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Example:

The ladder length is c=√(3²+4²)=5 m. Its angle to the ground satisfies tanθ=4/3, so θ≈53.1°. With two sides 6 and 8 enclosing 60°, c²=6²+8²-2×6×8 cos60°=52.

Right triangles and non-right triangles — original teaching diagram

Test a tempting shortcut

  • Label sides relative to the chosen angle. Pythagoras needs a right angle. A calculator angle mode error can produce a plausible but wrong result. Keep unrounded values for later steps.
  • When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Warn:

Pythagoras applies to every triangle, including triangles without a right angle. This claim is false. Explain which definition or assumption it violates.

Interpret a new situation

  • Use a plan or elevation for a three-dimensional problem before applying a triangle rule. Explain why the chosen triangle contains the required length or angle.
  • A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

  • Current first-assessment-2021 Analysis and Approaches SL. This is authored concept support; the full guide is needed to certify every objective.
  • Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

Key:

The side opposite the right angle in a right-angled triangle. Choose the relationship, show the method, check its assumptions and interpret the result.

词汇 训练
English 中文 拼音
hypotenuse/haɪˈpɒtənjuːs/ 斜边 xié biān
3.4

弧度制、恒等式与三角方程

How far does the rim travel?

  • A rotating wheel sweeps an arc. Radians let its angle connect directly to the distance travelled along the rim.
  • This lesson studies radian 弧度: The angle subtended by an arc whose length equals the radius.

Choose the mathematical structure

  • For θ in radians, arc length s=rθ and sector area A=r²θ/2. Use sin²θ+cos²θ=1 and angle identities to simplify or solve. List every solution in the stated interval.
  • State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.

Work through a checked case

$$s=r\theta,\qquad A=\frac12 r^2\theta,\qquad \sin^2\theta+\cos^2\theta=1$$
  • Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.

Example:

For r=4 and θ=π/3, s=4π/3 and A=8π/3. Solving sin x=1/2 on 0≤x<2π gives x=π/6 and 5π/6. For sin2x=1/2, solve first on 0≤2x<4π before dividing by 2.

Radians, identities and trigonometric equations — original teaching diagram

Test a tempting shortcut

  • A calculator's inverse sine gives one principal value. It does not give every solution. Dividing by sin x can discard solutions where sin x=0; check that case first.
  • When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.

Warn:

The principal inverse sine value is always the only solution in a full turn. This claim is false. Explain which definition or assumption it violates.

Interpret a new situation

  • Choose an identity that reduces the equation to one trigonometric function. Keep exact familiar angles where possible, and check each solution against the original equation.
  • A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.

Use this in your course

  • Current first-assessment-2021 Analysis and Approaches SL. This is authored concept support; the full guide is needed to certify every objective.
  • Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.

Key:

The angle subtended by an arc whose length equals the radius. Choose the relationship, show the method, check its assumptions and interpret the result.

词汇 训练
English 中文 拼音
radian/ˈreɪdɪən/ 弧度 hú dù

该知识点的互动课程

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