加法定理と幾何学的証明
導入済み| English | 意味 |
|---|---|
| compound angle/ˈkɒmpaʊnd ˈæŋɡl/ | 合成角 |
Two unit-radius points define a chord. Can its length explain why cosine of a difference is not the difference of two cosines?
- Two unit-radius points define a chord. Can its length explain why cosine of a difference is not the difference of two cosines?
- This lesson studies compound angle 复合角: An angle expressed as the sum or difference of two angles.
Choose the mathematical structure
- sin(A±B)=sinA cosB±cosA sinB. cos(A±B)=cosA cosB∓sinA sinB: the cosine sign changes. tan(A±B)=(tanA±tanB)/(1∓tanA tanB) where the tangents and denominator are defined. Derive tangent by dividing the corresponding sine and cosine formulas; do not extend a quotient through a zero denominator.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
なぜ cos(A+B)=cosA cosB - sinA sinB であり、sin(A+B) はプラスになるのか？
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
A=B=45° で検証：cos 90°=0 および cosAcosB-sinAsinB=1/2-1/2=0。ただし、プラスを適用すると 1 となる。余弦の符号反転は、正確に合成幾何学によって生成される値である。
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Let P=(cosA,sinA) and Q=(cosB,sinB) on the unit circle. Coordinate distance gives PQ²=2−2(cosA cosB+sinA sinB). The cosine rule on the two unit radii gives PQ²=2−2cos(A−B), hence the cosine difference formula. Replace B by −B, using cosine even/sine odd, to obtain cosine addition. Apply cosine difference to (π/2−A) and B to obtain sin(A+B); then change B’s sign for sine subtraction. Exact evaluations: sin75°=sin(45°+30°)=(√6+√2)/4; cos75°=(√6−√2)/4; sin15°=(√6−√2)/4, cos15°=(√6+√2)/4 and tan15°=2−√3.
加法定理と幾何学的証明
sin(A±B)=sinA cosB±cosA sinB
幾何学的条件または係数条件から角の公式が導かれることを説明せよ。
sin(π/12) の値を小数で求めよ。
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
sin(π/4−π/6)=(√6−√2)/4≈0.258819 を用いる。
Test a tempting shortcut
- sin(A+B) is not sinA+sinB. The cosine rule chord uses the smaller included angle when necessary, but its cosine is still cos(A−B). Coordinate distance and the resulting identities hold for all signed angles; the initial diagram is one illustrative case. tan(45°+45°) is undefined because 1−tan45°tan45°=0.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Cosine of a sum always equals the sum of the two cosine values. This claim is false. Explain which definition or assumption it violates.
cos(π/12) の小数値を求めよ。
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
cos(π/4−π/6)=(√6+√2)/4≈0.965926 を用いる。
和の余弦は、必ずしも二つの余弦値の和に等しいわけではない。
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
sin(A+B) は sinA+sinB に等しくない。必要に応じて余弦定理の弦には较小的な内包角を用いるが、その余弦は依然として cos(A−B) である。座標距離と導かれる恒等式はすべての符号付き角度について成立する；初期図示はその一例である。tan(45°+45°) は定義されない。なぜなら 1−tan45°tan45°=0 だからである。
Interpret a new situation
- Choose familiar angles whose sum or difference is the target. Keep exact roots and rationalise a tangent quotient if needed. In a geometric proof, state both expressions for the same chord rather than assuming the desired identity. Check the denominator before applying a tangent addition/subtraction formula.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
tan(π/12) の小数値を求めよ。
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
tan(π/4−π/6)=2−√3≈0.267949 を用いる。
各正確な値を、合成角計算と対応させよ。
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
特殊三角形の値を用い、電卓を使わずに合成公式で 15° と 75° の正確な値を導き出す。
Use this in your course
- 7357 · A-level · E. Match the target tier and specification before assigning extensions.
- 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.
An angle expressed as the sum or difference of two angles. Choose the relationship, show the method, check its assumptions and interpret the result.