Compound-angle formulae and geometric proofs
| English | 中文 | Pinyin |
|---|---|---|
| compound angle/ˈkɒmpaʊnd ˈæŋɡl/ | 复合角 | fù hé jiǎo |
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.
Which description correctly defines compound angle?
An angle expressed as the sum or difference of two angles.
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.
Compound-angle formulae and geometric proofs
sin(A±B)=sinA cosB±cosA sinB
Explain how the angle formula follows from the geometric or coefficient conditions.
Find sin(π/12) as a decimal.
Use sin(π/4−π/6)=(√6−√2)/4≈0.258819.
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.
Find cos(π/12) as a decimal.
Use cos(π/4−π/6)=(√6+√2)/4≈0.965926.
Cosine of a sum always equals the sum of the two cosine values.
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.
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.
Find tan(π/12) as a decimal.
Use tan(π/4−π/6)=2−√3≈0.267949.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
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.