Sine rule, cosine rule and triangle area · Higher
도입됨| English | 의미 |
|---|---|
| included angle/ɪnˈkluːdɪd ˈæŋɡl/ | 포함된 각 |
A triangular plot has no right angle. Two known sides and their included angle can determine the remaining side without inventing a perpendicular side length.
- A triangular plot has no right angle. Two known sides and their included angle can determine the remaining side without inventing a perpendicular side length.
- This lesson studies included angle 夹角: The angle between the two named sides.
Choose the mathematical structure
- Sine rule pairs opposite sides/angles: a/sinA=b/sinB=c/sinC. Cosine rule a²=b²+c²-2bc cosA uses the angle opposite a. Area is ab sinC/2 when C lies between a and b. Choose the rule from the known information. An inverse sine may give an acute angle and an obtuse supplement; check the angle sum and supplied sides before accepting either.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Sides 6 and 8 include a 60° angle. Why does the area formula use sin 60°?
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
The formula A=ab sin C/2 treats one side as base and derives the height from the other side times the sine of the angle BETWEEN them. A non-included angle gives a wrong height.
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.
With sides 6 and 8 enclosing 60°, c²=36+64-96×(1/2)=52, hence c=2√13. Its area is (1/2)×6×8×sin60°=12√3. For a=4 opposite A=30° and B=45°, b=4 sin45°/sin30°=4√2. If sides a=7,b=5,c=6, cosA=(25+36-49)/(2×5×6)=1/5, so A≈78.5°. To find an angle from area 12 with enclosing sides 6 and 8, sinC=24/48=1/2; C could be 30° or 150° until the remaining data selects a shape. Label opposite pairs and check triangle inequalities to reject impossible side combinations.
Sine rule, cosine rule and triangle area
Sine rule pairs opposite sides/angles: a/sinA=b/sinB=c/sinC
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
For sides 6,8 and included 60°, find the square of the third side.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
36+64-96×cos60°=52.
Test a tempting shortcut
- Do not pair a side with its adjacent angle in the sine rule. The area angle must be included. A calculator’s first inverse-sine answer need not be the only possible triangle.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The sine rule pairs each side with any angle in the triangle. This claim is false. Explain which definition or assumption it violates.
For sides 7,5,6, find cosine of the angle opposite 7.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
(25+36-49)/(2×5×6)=12/60.
The sine rule pairs each side with any angle in the triangle.
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
Do not pair a side with its adjacent angle in the sine rule. The area angle must be included. A calculator’s first inverse-sine answer need not be the only possible triangle.
Interpret a new situation
- AQA G22/G23 Higher includes unknown sides/angles and areas of general triangles. Write the chosen rule before substitution and state any second possible configuration.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
For area 12 and enclosing sides 6,8, find sine of the included angle.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
sinC=2×12/(6×8)=1/2.
Match each triangle rule to the information it needs.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
Choose the rule from what is known: opposite pairs call for the sine rule, an included angle calls for the cosine rule or the area formula.
Use this in your course
- 8300 · Higher · 3.4. 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.
The angle between the two named sides. Choose the relationship, show the method, check its assumptions and interpret the result.