The three sides and three angles of a triangle are tied together, so knowing just a few of them lets you work out the rest.
The law of sines says that each side divided by the sine of the opposite angle equals the diameter of the circumcircle, 2R, and the law of cosines, c² = a² + b² − 2ab cos C, turns into the Pythagorean theorem when the included angle is a right angle.
Add the area formula S = (1/2)ab sin C, and distances and angles that cannot be measured directly, as in surveying or navigation, become reachable.
The first picture draws the diameter BA′ to show why a = 2R sin A; in the second, the third side c is recalculated each time the two sides or the included angle change.
The law of sines: the diameter of the circumcircle
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Triangle ABC on a circumcircle of radius 5, with the diameter BA′; ∠A′ = ∠A and ∠BCA′ = 90°
50°
Law of Sines
asin A = bsin B = csin C = 2R
Each side divided by the sine of the opposite angle equals the diameter of the circumcircle
💡 Why a = 2R sin A
①Draw the diameter BA′ through the center; as an angle in a semicircle, ∠BCA′ = 90°
②Both stand on the same arc BC, so ∠A′ = ∠A
③In the right triangle A′BC, BC = BA′ × sin A′, which gives a = 2R sin A
④If A is obtuse, ∠A′ = 180° − A instead, but the sine is the same, so the same formula holds
The law of cosines: Pythagoras for every triangle
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A triangle in which the third side c is found from the sides a, b and the included angle C
4
3
60°
Law of Cosines
c² = a² + b² − 2ab cos C
When C = 90°, cos C = 0 and it becomes the Pythagorean theorem
Area of a Triangle
Area from two sides and the included angle
S = 12 ab sin C
With side a as the base, the height is b sin C, so the area is (1/2) × a × b sin C
Heron's Formula
S = √(s(s-a)(s-b)(s-c)), s = a+b+c2
Used when only the three sides are known
📐 Which formula to use
①Two sides and the included angle: S = (1/2)ab sin C
②Only the three sides: find the cosine of one angle with the law of cosines and then the area, or use Heron's formula
③One side and the two angles at its ends: find another side with the law of sines, then the area
In everyday life
🌍 Where Trig Is Used
①Surveying: measuring the height of a building or mountain you cannot climb from angles and distances
②Navigation: finding a ship's position and distance from bearings taken at two points
③Physics: splitting a force into two directions or combining forces
④Architecture: setting roof pitches and the lengths of structural members
Summary
Law of Sines
asin A = bsin B = csin C = 2R
side ÷ sine of the opposite angle = diameter of the circumcircle
Law of Cosines
c² = a² + b² − 2ab cos C
The Pythagorean theorem extended to every triangle
🎯 Exam Points
①Law of sines: two angles and a side, or finding the circumradius
②Law of cosines: two sides and the included angle, or all three sides
③Area S = (1/2)ab sin C
④When C = 90°, cos C = 0 and it becomes the Pythagorean theorem
⑤Find the circumradius R from a/sin A = 2R
Worked Examples & Past Exam
Example 1
In triangle ABC, a = 8 and sin A = 1/2. Find the circumradius R.
1
By the law of sines, a/sin A equals the diameter of the circumcircle, 2R.
2R = asin A = 81/2
2
8 divided by 1/2 is 16, so 2R = 16.
2R = 16, R = 8
▸ R = 8
sin A = 1/2 means A is 30° or 150°, but either way the circumradius is 8. The law of sines holds whether the angle is acute or obtuse.
Example 2
With sides a = 5, b = 3 and included angle C = 60°, use the law of cosines to find c.
1
Two sides and the included angle are known, so use the law of cosines.
c² = 5² + 3² - 2(5)(3) cos 60°
2
cos 60° = 1/2, so subtract 2 × 5 × 3 × 1/2 = 15.
c² = 25 + 9 - 30 × 12 = 19, c = √19
▸ c = √19
If C is less than 90°, c² is smaller than a² + b²; if it is more than 90°, c² is larger. The sign of cos C makes the difference.
2023 CSAT Math type, adapted
Find the area of a triangle with sides a = 4, b = 6 and included angle C = 30°.
①6
②12
③6√3
④3
⑤12√3
▸ ① 6
1
Put the sides 4 and 6 and the included angle 30° into the area formula S = (1/2)ab sin C.