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Lesson 15 — Law of sines and law of cosines

Solving general (non-right) triangles. Applications in surveying, navigation, and physics.

Used in: 1.º ano do EM (15 anos) · Math II japonês (cap. 図形と計量) · Trigonometry — US precalc

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R, \quad c^2 = a^2 + b^2 - 2ab\cos C

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Rigorous notation, full derivation, hypotheses

Proofs and use

Law of sines

what this means · Holds for any triangle (acute, obtuse, right). R is the radius of the circumscribed circle.

Proof (for an acute triangle): construct the altitude $h$ from vertex $C$ to side $\overline{AB}$ . Then $h = b \sin A = a \sin B$ . Therefore $a/\sin A = b/\sin B$ . Same argument for $c$ . ∎

Special case (right angle at $C$ ): $\sin C = 1$ , so $c = 2R$ — the hypotenuse is the diameter of the circumscribed circle. Thales' theorem (geometric).

Law of cosines

what this means · Generalizes Pythagoras. When C = 90°, cos C = 0 and we recover c² = a² + b².

Proof: by the dot product of vectors $\vec{CB} - \vec{CA} = \vec{AB}$ : $|\vec{AB}|^2 = |\vec{CB}|^2 + |\vec{CA}|^2 - 2 \vec{CB} \cdot \vec{CA}$

Since $\vec{CB} \cdot \vec{CA} = |\vec{CB}||\vec{CA}|\cos C = ab \cos C$ , we obtain $c^2 = a^2 + b^2 - 2ab \cos C$ . ∎

When to use each law

You have	You want	Use
2 angles + 1 side (AAS, ASA)	the other sides	Law of sines
2 sides + angle opposite to one (SSA)	remaining (ambiguous!)	Law of sines
2 sides + angle between them (SAS)	third side	Law of cosines
3 sides (SSS)	any angle	Law of cosines inverted

Ambiguous case (SSA)

Given $a$ , $b$ , and $A$ (angle opposite to $a$ ): there may be 0, 1, or 2 triangles. Decision:

If $a \geq b$ : 1 triangle.
If $a < b \sin A$ : 0 triangles (geometrically impossible).
If $b \sin A < a < b$ : 2 triangles.

Exercise list

35 exercises · 8 with worked solution (25%)

Application 18Understanding 2Modeling 12Proof 3

Ex. 15.1Application
Triangle with $a = 8$ , $A = 30°$ , $B = 45°$ . Calculate $b$ .
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Ex. 15.2Application
Triangle with $a = 12$ , $A = 50°$ , $B = 70°$ . Calculate $b$ and $c$ .
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Ex. 15.3Application
Triangle with $a = 5$ , $b = 8$ , $A = 30°$ . How many triangles are possible?
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Ex. 15.4ApplicationAnswer key
Triangle with $b = 10$ , $B = 45°$ , $A = 60°$ . Calculate $a$ .
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Ex. 15.5ApplicationAnswer key
In a triangle $ABC$ , $A = 40°$ , $B = 80°$ , $a = 7$ . Calculate $C$ and $c$ .
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Ex. 15.6Application
Triangle with $a = 6$ , $A = 35°$ , $B = 50°$ . Calculate area.
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Ex. 15.7ApplicationAnswer key
Law of sines: $a/\sin 30° = c/\sin 90°$ . For $a = 4$ , calculate $c$ .
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Ex. 15.8Application
In a triangle, $a = 10$ , $b = 7$ , $A = 90°$ . Confirm with the law of sines.
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Ex. 15.9Application
Triangle: $A = 50°$ , $a = 12$ . Determine the radius of the circumscribed circle $R$ .
Solve onlineref: OpenStax A&T §10.1
Ex. 15.10Understanding
Show that in an equilateral triangle ( $A = B = C = 60°$ ), $a = b = c$ .
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Ex. 15.11Application
Triangle with $a = 5$ , $b = 7$ , $C = 60°$ . Calculate $c$ .
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Ex. 15.12Application
Triangle with $a = 8$ , $b = 6$ , $C = 90°$ . Calculate $c$ . (Recover Pythagoras.)
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Ex. 15.13Application
Triangle with $a = 4$ , $b = 3$ , $C = 120°$ . Calculate $c$ .
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Ex. 15.14Application
Triangle with $a = 5$ , $b = 6$ , $c = 7$ . Calculate $C$ .
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Ex. 15.15Application
Triangle with $a = 10$ , $b = 12$ , $c = 15$ . Determine the 3 angles.
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Ex. 15.16ApplicationAnswer key
In a triangle, $a = 12$ , $b = 8$ , $A = 80°$ . Use the law of sines for $B$ and then calculate $c$ .
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Ex. 15.17Application
Triangle $ABC$ : $a = 4$ , $b = 5$ , $c = 6$ . Calculate area using Heron's formula.
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Ex. 15.18Application
In an equilateral triangle of side $\ell$ , show via the law of cosines that each angle is $60°$ .
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Ex. 15.19Application
Triangle with sides $7, 24, 25$ . Verify it is a right triangle via the law of cosines.
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Ex. 15.20Understanding
When $C \to 0$ , what does the law of cosines $c^2 = a^2 + b^2 - 2ab\cos C$ tend to? Interpret geometrically.
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Ex. 15.21Modeling
You walk 5 km east, then turn $60°$ north and walk 3 more km. Distance from the origin?
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Ex. 15.22Modeling
A ship leaves port, sails 12 km northwest, then 8 km northeast. Distance from origin?
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Ex. 15.23Modeling
A drone observes two points $A$ and $B$ on the ground at angles of $50°$ and $70°$ . Drone at 200 m altitude. Calculate distance $AB$ .
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Ex. 15.24ModelingAnswer key
Two sides of a triangular plot measure 80 m and 100 m, forming a $75°$ angle. Length of the third side?
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Ex. 15.25ModelingAnswer key
In a soccer field, an attacker shoots from the position seeing the 6-meter goal at a $20°$ angle from position $A$ (dist. to goal = 30 m). Distance from goal to attacker from $A$ ? (Goal geometry and angle.)
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Ex. 15.26Modeling
Surveying: you need to measure the distance between two points $A$ and $B$ separated by a river. You're at $C$ , with $\hat{ACB} = 60°$ , $AC = 50$ m, $BC = 70$ m. Distance $AB$ ?
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Ex. 15.27ModelingAnswer key
Astronomy: stellar parallax of a star measures an angle of $\pi/(360 \cdot 60 \cdot 60)$ rad (1 arc-second) from one side to the other of Earth's orbit. What is the distance to the star in AU? (Ans: 206,265 AU = 1 parsec.)
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Ex. 15.28Modeling
An irrigation triangle has sides 100m, 120m, 80m. Area?
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Ex. 15.29Modeling
Inverse kinematics: a robotic arm with 2 segments $\ell_1 = 30$ cm, $\ell_2 = 25$ cm needs to reach a point at distance $r = 40$ cm. Angle between segments?
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Ex. 15.30ModelingAnswer key
Resultant velocity of a boat at $5$ km/h in a river with $3$ km/h perpendicular current: magnitude and angle?
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Ex. 15.31Modeling
A plane travels at 500 km/h on heading $60°$ NE. Wind blows at 100 km/h from the east. Resultant velocity?
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Ex. 15.32Modeling
In 2D GPS, two satellites at $(0, 100)$ and $(50, 80)$ km see you at angles $30°$ and $45°$ — describe (do not calculate) the triangulation.
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Ex. 15.33Proof
Prove the law of sines for an acute triangle, using the altitude from vertex $C$ .
Ex. 15.34Proof
Prove the law of cosines for any triangle, using the dot product.
Ex. 15.35Proof
Prove Heron's formula using the law of cosines + area = (1/2)ab sin C.

Sources for this lesson

Algebra and Trigonometry — Jay Abramson et al. (OpenStax) · 2022, 2nd ed · EN · CC-BY · §10.1-10.2: laws of sines and cosines. Primary source.
Precalculus / College Algebra / Trigonometry — Stitz, Zeager · 2013, v3 · EN · CC-BY-NC-SA · §11.2-11.3: non-right triangles.
College Trigonometry — Stitz, Zeager · 2013 · EN · CC-BY-NC-SA · ch. 11: applications.
University Physics (Volume 1) — OpenStax · 2016 · EN · CC-BY · ch. 2: vectors and vector addition. Source for block C.