In a triangle the sides and angles are linked, so a few known parts reveal the rest.
The law of sines makes each side over the sine of its opposite angle equal the circumdiameter 2R, while the law of cosines c² = a² + b² − 2ab cos C is a generalized Pythagoras that reduces to it at a right angle.
With the area formula S = (1/2)ab sin C you can find lengths and angles no ruler can reach, as in surveying or navigation.
Adjust the vertex, sides, and angle with the sliders to see how the two laws fix a triangle.
Law of Sines — Circumscribed Circle
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A triangle on its circumcircle and the law of sines
50°
Law of Sines
asin A = bsin B = csin C = 2R
Each side over the sine of the opposite angle equals the circumdiameter
💡 Key Idea
①'Side / sine of opposite angle' = circumdiameter (2R)
②Knowing 2 angles + 1 side gives the rest
③Area: S = (1/2)ab sin C
Law of Cosines — Generalized Pythagoras
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Find the remaining side from two sides and the included angle
4
3
60°
Law of Cosines
c² = a² + b² - 2ab cos C
When C = 90° it reduces to the Pythagorean theorem
Area of a Triangle
Area (two sides + included angle)
S = 12 ab sin C
Two sides and the included angle
Heron's Formula
S = √(s(s-a)(s-b)(s-c)), s = a+b+c2
Compute area from three sides only
📐 Which Formula to Use?
①Two sides + included angle → S = (1/2)ab sin C
②Three sides only → Heron
③One side + two adjacent angles → law of sines first, then area
Real-life Applications
🌍 Where Trig Is Used
①Surveying: height of buildings/mountains
②Navigation: distance between two points
③Physics: force decomposition, vector composition
④Architecture: roof pitch, structural design
Wrap-up
Law of Sines
asin A = bsin B = csin C = 2R
side/sin = circumdiameter
Law of Cosines
c² = a² + b² - 2ab cos C
Generalization of Pythagoras
🎯 Exam Points
①Law of sines: angle-side correspondence (ASA, AAS)
②Law of cosines: SAS or SSS
③Area: S = (1/2)ab sin C
④C = 90° → cos C = 0 → Pythagoras
⑤Law of sines also gives circumradius R
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 = 2R.
2R = asin A = 81/2
2
Compute to find R.
2R = 16, R = 8
▸ R = 8
Use the law of sines a/sin A = 2R to find the circumradius.
Example 2
With sides a = 5, b = 3 and included angle C = 60°, use the law of cosines to find c.
1
Substitute into the law of cosines c² = a² + b² − 2ab cos C.
c² = 5² + 3² - 2(5)(3) cos 60°
2
Substitute cos 60° = 1/2 and compute.
c² = 25 + 9 - 30 × 12 = 19, c = √19
▸ c = √19
Given two sides and the included angle, the law of cosines finds the third side.
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
Substitute into the area formula S = (1/2)ab sin C.