Non-right-angled triangles
Section: Trigonometry | Syllabus: Cambridge IGCSE Mathematics (0580)
The Sine Rule
Extended Only
The sine rule connects the sides and angles of any triangle, not just right-angled ones. It is used when a pair of an angle and its opposite side is known, along with one other side or angle.
- The sine rule: (a/sin A)=(b/sin B)=(c/sin C), where each side is paired with the angle opposite it
- To find a missing side, use the rule with two sides and their opposite angles; to find a missing angle, rearrange to put the sine terms on top
- When finding an angle using the sine rule, check whether the answer could be obtuse instead of acute - the "ambiguous case" - since sin(x) and sin(180-x) give the same value
Worked Example: Finding a Side Using the Sine Rule
- Question: In triangle ABC, angle A = 48°, angle B = 67°, and side a (opposite A) is 15 cm. Find side b (opposite B).
- Step 1: (a/sin A)=(b/sin B)
- Step 2: (15/48)=(b/67)
- Step 3: b=(15×67/48)18.6
- Answer: 18.6 cm (3 s.f.)
Worked Example: Finding an Angle Using the Sine Rule
- Question: In triangle PQR, side p = 22 cm (opposite angle P), side q = 18 cm (opposite angle Q), and angle Q = 40°. Find angle P.
- Step 1: sin P=(p Q/q)
- Step 2: sin P=(22×40/18)0.7857
- Step 3: P=sin^-1(0.7857)51.8°
- Answer: P ≈ 51.8° (3 s.f.) - the acute solution, since the triangle's angles must sum to 180° with the given 40°
Common Mistakes
MistakePairing a side with the wrong angle, e.g. using side a with angle B instead of angle A
Fixeach side in the sine rule must be paired with the angle directly opposite it - check the triangle carefully before substituting
MistakeForgetting to consider whether an angle found using the sine rule could be obtuse instead of acute (the ambiguous case)
Fixcheck whether the obtuse alternative (180° minus the calculator's answer) is also possible, based on the other angles already known in the triangle
The Cosine Rule
Extended Only
The cosine rule connects all three sides of a triangle with one angle. It is used when the sine rule cannot be applied - typically when two sides and the angle between them are known, or when all three sides are known.
- To find a missing side: a^2=b^2+c^2-2bccos A, where A is the angle between the two known sides b and c
- To find a missing angle when all three sides are known, rearrange to: cos A=(b^2+c^2-a^2/2bc)
- The cosine rule works for any triangle, and does not have an ambiguous case like the sine rule does
Worked Example: Finding a Side Using the Cosine Rule
- Question: In triangle XYZ, side x = 310 m, side y = 520 m, and the angle between them, angle Z, is 112°. Find side z.
- Step 1: z^2=x^2+y^2-2xycos Z
- Step 2: z^2=310^2+520^2-2310520(112)
- Step 3: z^2=96\,100+270\,400-(-120\,801.7)487\,301.7
- Step 4: z=√(487\,301.7)698
- Answer: 698 m (3 s.f.)
Common Mistakes
MistakeForgetting that cosine is negative for obtuse angles, and mishandling the sign when subtracting a negative value
Fixwork out cos(A) fully first (it will be negative for an obtuse angle), then carefully subtract this negative value, which actually increases the total
MistakeUsing the cosine rule when the sine rule would be simpler, based on which information is actually known
Fixthe cosine rule needs either all three sides, or two sides and the angle between them - if instead an angle and its opposite side are both known, the sine rule is usually the better choice
Area of a Triangle Using ½ab sin C
Extended Only
The area of any triangle can be found using two sides and the angle between them, without needing to know the perpendicular height.
- Area =(1/2)× a× b C, where a and b are two sides and C is the angle between them (the included angle)
- This formula is especially useful when the height of the triangle isn't given directly
- The two sides used must be the ones that meet at the angle being used - not any other pair
Worked Example: Finding the Area of a Triangle Using Two Sides and the Included Angle
- Question: Triangle DEF has DE = 380 m, DF = 250 m, and the angle between them, angle D, is 63°. Find the area of the triangle.
- Step 1: (1/2)380250(63)
- Step 2: =47\,500(63)
- Step 3: 42\,322.8
- Answer: 42 300 m² (3 s.f.)
Common Mistakes
MistakeUsing two sides that don't actually meet at the given angle, instead of the correct included pair
Fixthe angle in the formula must be the one directly between the two chosen sides - check the diagram carefully
MistakeForgetting the ½ at the front of the formula
Fixthe formula is always ½ab sin C - dropping the ½ doubles the answer incorrectly
Choosing Which Rule to Use
Extended Only
With right-angle trigonometry, the sine rule, and the cosine rule all available, choosing the correct method for a given triangle problem is an important first step.
- If the triangle has a right angle, use ordinary right-angle trigonometry (SOH CAH TOA) - it's simpler than the sine or cosine rule
- Use the sine rule when a matching angle-side opposite pair is known, along with one more side or angle
- Use the cosine rule when two sides and the included angle are known (to find the third side), or when all three sides are known (to find an angle)
- Use the ½ab sin C formula specifically for area, whenever two sides and the included angle are known
Worked Example: Deciding Which Rule to Apply
- Question: A triangle has sides of 14 cm, 19 cm and 23 cm, with no right angle and no given angles. Explain which rule should be used to find one of its angles, and why.
- Step 1: No angle is given, only three sides - the sine rule cannot be used, since it always needs at least one angle-side opposite pair
- Step 2: Since all three sides are known, the cosine rule (rearranged to find an angle) is the correct choice
- Answer: use the cosine rule, since all three sides are known but no angle is given
Common Mistakes
MistakeTrying to force the sine rule onto a triangle where no angle-side opposite pair is actually known
Fixcheck what information is actually given first - if only three sides are known, the cosine rule (rearranged) is the only option
MistakeUsing the cosine rule for a triangle where right-angle trigonometry would be simpler and sufficient
Fixalways check for a right angle first - if one exists, SOH CAH TOA is usually quicker than either the sine or cosine rule
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