Sine Rule, Cosine Rule and Area of a Triangle
Choose the sine rule, cosine rule or triangle area formula, handle obtuse angles and the ambiguous case, and work with exact or rounded answers.
Choose the rule from the information
In any triangle, side a is opposite angle A, and similarly for b,B and c,C.
Sine rule
\frac a{\sin A}=\frac b{\sin B}=\frac c{\sin C}
Use it when you have a known opposite side–angle pair.
Cosine rule
a^2=b^2+c^2-2bc\cos A
Use two sides and their included angle to find the third side, or three sides to find an angle.
Area of a triangle
\frac12bc\sin A
Use the angle between the two known sides.
Set the calculator to degrees. Keep full precision until the final answer. In a triangle, each angle is between 0° and 180°, and all three add to 180°.
Worked example 1
In triangle ABC, A=38^\circ, B=67^\circ and a=9 cm. Find b to 3 significant figures.
- \frac b{\sin67^\circ}=\frac9{\sin38^\circ}, so b=\frac{9\sin67^\circ}{\sin38^\circ}
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Your turn. Evaluate b to 3 significant figures, in cm.
b=13.4563\ldots cm, which rounds to 13.5 cm.
- Answer: b=13.5 cm (3 s.f.). It is longer than a, as it is opposite the larger angle.
Worked example 2
In triangle ABC, A=42^\circ, a=11 cm and b=8 cm. Find B to 1 decimal place and explain why there is only one possible triangle.
- Use \frac{\sin B}{8}=\frac{\sin42^\circ}{11}, so \sin B=\frac{8\sin42^\circ}{11}
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Your turn. Calculate the acute value of B, to 1 decimal place.
B=29.1200\ldots^\circ
- The other angle with the same sine is 180-29.1200\ldots=150.8799\ldots^\circ. Adding 42^\circ would exceed 180^\circ, so it cannot fit in this triangle.
- Answer: B=29.1^\circ (1 d.p.), with only one valid triangle.
Check for a second triangle
When two sides and a non-included angle are known, the sine rule can give two triangles. If \sin B=k with 0<k<1, test both:
B=\sin^{-1}k
B=180^\circ-\sin^{-1}k
Reject any candidate leaving no positive third angle.
Worked example 3
In triangle ABC, A=32^\circ, a=7 cm and b=10 cm. Find both possible values of B and the corresponding values of C, to 1 decimal place.
- \sin B=\frac{10\sin32^\circ}{7} gives B_1=49.2028\ldots^\circ and B_2=180^\circ-B_1=130.7971\ldots^\circ
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Your turn. Give the obtuse value of B to 1 decimal place.
B_2=130.8^\circ (1 d.p.).
- Use C=180^\circ-32^\circ-B with each unrounded value. Both resulting angles are positive.
- Answer: (B,C)=(49.2^\circ,98.8^\circ) or (130.8^\circ,17.2^\circ), all to 1 d.p. Keep the angles in matching pairs.
Worked example 4
Two sides of a triangle are 7 cm and 11 cm with included angle 120^\circ. Find the third side exactly and to 3 significant figures.
- a^2=7^2+11^2-2(7)(11)\cos120^\circ. Since \cos120^\circ=-\frac12, the last term increases the result.
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Your turn. Calculate a^2=49+121+77
a^2=247
- Answer: a=\sqrt{247} cm, or 15.7 cm (3 s.f.). Take the square root after using the cosine rule.
Worked example 5
A triangle has sides 7 cm, 8 cm and 12 cm. Find the angle opposite the 12 cm side, to 1 decimal place.
- Rearrange the cosine rule: \cos A=\frac{7^2+8^2-12^2}{2(7)(8)}=\frac{-31}{112}
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Your turn. Calculate A=\cos^{-1}(-31/112) to 1 decimal place.
A=106.0684\ldots^\circ
- Answer: 106.1^\circ (1 d.p.). A negative cosine gives an obtuse triangle angle.
Worked example 6
Find the exact area of a triangle with sides 8 cm and 13 cm enclosing an angle of 30^\circ.
- Use \text{area}=\frac12(8)(13)\sin30^\circ and \sin30^\circ=\frac12
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Your turn. Find the area in cm².
\frac12(8)(13)(\frac12)=26
- Answer: 26 cm². The angle used must lie between the two supplied sides.
Worked example 7
A triangle has sides 6 cm and 10 cm and area 15\sqrt3 cm². Find both possible included angles and the corresponding third-side lengths exactly.
- 15\sqrt3=\frac12(6)(10)\sin C, so \sin C=\sqrt3/2. The angle lies between 0^\circ and 180^\circ
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Your turn. The acute solution is 60^\circ. What is the obtuse solution, in degrees?
C=60^\circ or 120^\circ
- For C=60^\circ, c^2=36+100-120(\frac12)=76, so c=2\sqrt{19} cm. For C=120^\circ, c^2=36+100-120(-\frac12)=196, so c=14 cm.
- Answer: 60^\circ with third side 2\sqrt{19} cm, or 120^\circ with third side 14 cm.
Common mistakes
- Pairing a side with an adjacent angle. The sine rule pairs each side with its opposite angle.
- Using the wrong angle between two sides. The area formula and cosine rule need the included angle for those two sides.
- Missing the second sine-rule angle. Test the supplementary angle and check the triangle’s angle sum.
- Rejecting a negative cosine value. A cosine between -1 and 0 gives an obtuse angle.
- Rounding early or leaving a side squared. Keep full precision, then take the positive square root when finding a side.