Sine rule, cosine rule and triangle area
Введено| English | Значение |
|---|---|
| included angle/ɪnˈkluːdɪd ˈæŋɡl/ | включённый угол |
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.
- This lesson studies included angle 夹角: The angle between the two named sides.
The method
- 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.
Sides 6 and 8 include a 60° angle. Why does the area formula use sin 60°?
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
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.
A worked case
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.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
36+64-96×cos60°=52.
Where it goes wrong
- 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.
Not true: The sine rule pairs each side with any angle in the triangle. Each side is paired with the angle opposite it: a/sin A=b/sin B.
For sides 7,5,6, find cosine of the angle opposite 7.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
(25+36-49)/(2×5×6)=12/60.
The sine rule pairs each side with any angle in the triangle.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
Each side is paired with the angle opposite it: a/sin A=b/sin B.
Applying it
- 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.
For area 12 and enclosing sides 6,8, find sine of the included angle.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
sinC=2×12/(6×8)=1/2.
Match each triangle rule to the information it needs.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
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.
In your course
- AQA A-level Mathematics (7357), specification section E.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Included angle: The angle between the two named sides.
- Sine rule pairs opposite sides/angles: a/sinA=b/sinB=c/sinC.
- Watch for: do not pair a side with its adjacent angle in the sine rule.