Sine rule, cosine rule and triangle area
| English | 中文 | Pinyin |
|---|---|---|
| included angle/ɪnˈkluːdɪd ˈæŋɡl/ | 夹角 | jiā jiǎo |
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.
Which description correctly defines included angle?
The angle between the two named sides.
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.
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.
(25+36-49)/(2×5×6)=12/60.
The sine rule pairs each side with any angle in the triangle.
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 7357 E1 includes sine/cosine rules and general-triangle area. Choose the rule from the known opposite pairs or included angle; preserve exact values and test all feasible triangle configurations. The dedicated ambiguous-case lesson develops the non-included-angle case.
- 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.
sinC=2×12/(6×8)=1/2.
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.
The angle between the two named sides. Choose the relationship, show the method, check its assumptions and interpret the result.