Proof of the law of sines from the law of cosines

Builds on Triangle Angle at a point Collinear points Euclidean distance Length of a vector Dot product Law of cosines Linear independence Cauchy–Schwarz inequality Sine and cosine Pythagorean trigonometric identity

Proof

We compute each sine from the three sides and find that the three quotients are given by one symmetric formula.

The vertices of the triangle are not collinear, so they are distinct. Thus $a, b, c > 0$ , and the three angles are defined.
We apply the law of cosines with the vertices renamed, so that $A$ plays the role of $C$ . It gives $a^2 = b^2 + c^2 - 2bc \cos \alpha$ . So $\cos \alpha = \dfrac{b^2 + c^2 - a^2}{2bc}$ .
In the same way, $\cos \beta = \dfrac{c^2 + a^2 - b^2}{2ca}$ and $\cos \gamma = \dfrac{a^2 + b^2 - c^2}{2ab}$ .
Let $Q = 4b^2c^2 - (b^2 + c^2 - a^2)^2$ . Two differences of squares give $Q = \left( a^2 - (b - c)^2 \right) \left( (b + c)^2 - a^2 \right) = (a - b + c)(a + b - c)(b + c - a)(a + b + c)$ .
A permutation of $a$ , $b$ , $c$ only permutes the four factors in step 4. So step 4 with the letters renamed gives $Q = 4c^2a^2 - (c^2 + a^2 - b^2)^2 = 4a^2b^2 - (a^2 + b^2 - c^2)^2$ .
Let $x = B - A$ and $y = C - A$ . With the length and the dot product, $\norm{x} = c$ , $\norm{y} = b$ and $x \cdot y = bc \cos \alpha = \frac{1}{2}(b^2 + c^2 - a^2)$ . So $Q = 4 \left( \norm{x}^2 \norm{y}^2 - (x \cdot y)^2 \right)$ .
Every difference of two vertices lies in the span of $x$ and $y$ . Since the vertices are not collinear, this span has dimension $2$ . So $x$ and $y$ are linearly independent.
By the Cauchy–Schwarz inequality and its case of equality, $(x \cdot y)^2 < \norm{x}^2 \norm{y}^2$ . So $Q > 0$ . Let $R = \sqrt{Q} > 0$ .
The sine is nonnegative on $[0, \pi]$ , and $\alpha \in [0, \pi]$ . By the identity $\sin^2 + \cos^2 = 1$ and step 2, $\sin^2 \alpha = 1 - \cos^2 \alpha = \dfrac{Q}{4b^2c^2}$ . So $\sin \alpha = \dfrac{R}{2bc} > 0$ .
In the same way, steps 3 and 5 give $\sin \beta = \dfrac{R}{2ca} > 0$ and $\sin \gamma = \dfrac{R}{2ab} > 0$ . So the three quotients are defined.
By steps 9 and 10, $\dfrac{a}{\sin \alpha} = \dfrac{2abc}{R}$ , $\dfrac{b}{\sin \beta} = \dfrac{2abc}{R}$ and $\dfrac{c}{\sin \gamma} = \dfrac{2abc}{R}$ .
Hence $\dfrac{a}{\sin \alpha} = \dfrac{b}{\sin \beta} = \dfrac{c}{\sin \gamma}$ .

Idea

The law of cosines gives each cosine from the sides, and then each sine. All three sines have the same numerator $\sqrt{Q}$ , a symmetric expression in the sides, which is four times the area. Dividing a side by the sine of the opposite angle then gives $2abc / \sqrt{Q}$ every time.

Reading path

To understand this page from the ground up, read these 261 pages in order. The foundations come first. The path takes the proofs Proof of the Cauchy–Hadamard theorem, Proof of the derivative of cosine by term-by-term differentiation, Proof of the Weierstrass M-test, Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the formula for the derivative of a polynomial, Proof of the commutativity of addition by induction, Proof of term-by-term differentiation, Proof of the epsilon-delta form of continuity, Proof of the power rule by induction, Proof that absolutely convergent series converge, Proof of the sum of a finite geometric series, Proof that convergent sequences are Cauchy, Proof that a complex number times its conjugate is the square of its modulus, Proof of the Cauchy–Schwarz inequality, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the chain rule with an auxiliary difference quotient, Proof of the Cauchy criterion for series, Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem, Proof of the equality of complex numbers, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the root test, Proof of the double negation law by a truth table, Proof that power series converge uniformly on smaller intervals inside the radius of convergence, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof of the uniqueness of limits, Proof of strong induction from ordinary induction, Proof that power series are differentiable inside the radius of convergence, Proof of the limit of a product of sequences, Proof that uniform convergence of derivatives allows term-by-term differentiation, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the existence of square roots of nonnegative reals, Proof of Pascal's rule from the recursive definition, Proof of the linearity of differentiation, Proof of the sequential characterization of limits, Proof that Cauchy sequences are bounded, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of the comparison test, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof of the factorial formula for binomial coefficients by induction, Proof of the binomial theorem by induction, Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms, Proof that differentiable functions are continuous, Proof of the epsilon-delta form of the limit of a function, Proof of the derivative of sine by term-by-term differentiation, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the Archimedean property in an Archimedean ordered field, Proof that a composition of continuous functions is continuous, Proof of the Pythagorean trigonometric identity by differentiation, Proof that a decreasing sequence bounded below converges, Proof of the term test, Proof that the reals are Archimedean from the completeness axiom, Proof of the ratio test, Proof that convergent sequences are bounded, Proof of the law of cosines with the dot product, Proof of the product rule, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof that a function with zero derivative on an open interval is constant, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof that a series of nonnegative terms converges iff its partial sums are bounded, Proof of the mean value theorem from Rolle's theorem, Proof of the constant multiple rule.

Angle at a point concept
Binomial coefficient definition
Collinear points concept
Complex numbers object
Complex numbers as pairs of reals concept
Construction of the reals from Cauchy sequences concept
Definition by recursion concept
Dot product concept
Element definition
Euclidean distance concept
Factorial definition
Group concept
Imaginary unit object
Inner product concept
Inner product space concept
Integers object
Length of a vector concept
Limit point of a set definition
Linear independence concept
Local maximum definition
Mathematical induction concept
Natural numbers object
Norm induced by an inner product concept
Ordered field definition
Peano axioms concept
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Sine and cosine concept
Triangle concept
Vector space definition
Proof of Pascal's rule from the recursive definition proof
Real and imaginary parts definition
Set definition
Ring definition
Proof of the law of cosines with the dot product proof
Ring of integers statement
Limit of a function definition
Local minimum definition
Successor function definition
Archimedean ordered field definition
Conjunction definition
Disjunction definition
Implication definition
Negation definition
Predicate definition
The reals form a field statement
Real sequence definition
Subsequence definition
Infimum definition
Supremum definition
Inner product definition
Linear combination definition
Norm definition
Pascal's rule statement
Complex conjugate definition
Ordered pair definition
Commutative ring definition
Continuity at a point definition
Derivative definition
One-sided limit definition
Biconditional definition
Law of cosines statement
Truth table concept
Addition of natural numbers definition
Modus ponens concept
Universal quantifier definition
The rationals form a field statement
Bounded sequence definition
Cauchy sequence definition
Geometric sequence definition
Limit of a sequence definition
Monotone sequence definition
Proof of the Archimedean property in an Archimedean ordered field proof
Sequence tending to infinity definition
Series definition
The successor function is injective statement
Zero is not a successor statement
Inner product space definition
Linear independence definition
Completeness axiom statement
Cartesian product definition
Proof of the factorial formula for binomial coefficients by induction proof
Continuous function definition
Proof of the equality of complex numbers proof
Differentiable function definition
One-sided derivative definition
Subset definition
Mathematical proof concept
Proof of the commutativity of addition by induction proof
Propositional logic concept
Archimedean property statement
Limit superior definition
Order on the natural numbers definition
Proof of the sum of a finite geometric series proof
Convergent sequence definition
Pointwise convergence definition
Sequential continuity definition
The rationals form an ordered field statement
The sequence r^n tends to infinity for r > 1 example
Partial sum definition
Power series definition
Norm induced by an inner product definition
Equality of complex numbers statement
Binary relation definition
Proof that a composition of continuous functions is continuous proof
Factorial formula for binomial coefficients statement
Tautology definition
Order on the reals definition
Commutativity of addition statement
Sum of a finite geometric series statement
Uniform convergence definition
Proof of strong induction from ordinary induction proof
Radius of convergence definition
Convergent series definition
Sine and cosine definition
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Proof of the binomial theorem by induction proof
Interval definition
Logical equivalence definition
Uniform convergence on a set definition
Strong induction statement
Series of functions definition
Sum of a series definition
Absolute convergence definition
Preorder definition
Binomial theorem statement
Proof of the epsilon-delta form of continuity proof
Proof of the epsilon-delta form of the limit of a function proof
Proof of the well-ordering principle from strong induction proof
Proof of the double negation law by a truth table proof
Partial order definition
Epsilon-delta definition of continuity statement
Epsilon-delta definition of a limit statement
Double negation law statement
Well-ordering principle statement
Total order definition
Composition of continuous functions is continuous statement
Proof by contradiction concept
Proof that every sequence has a monotone subsequence, by peaks proof
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
The reals form an ordered field statement
Absolute value definition
Proof of the approximation property of the supremum proof
Proof of the existence of square roots of nonnegative reals proof
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof of the epsilon-N definition of the limit of a sequence proof
Existence of square roots of nonnegative reals statement
Approximation property of the supremum statement
Triangle inequality statement
Absolute value is multiplicative statement
Square root definition
Epsilon-N definition of the limit of a sequence statement
Proof that the reals are Archimedean from the completeness axiom proof
Proof of the limit of a sum of sequences proof
Proof of the squeeze theorem for sequences proof
Proof of the uniqueness of limits proof
Proof that limits preserve non-strict inequalities proof
Modulus of a complex number definition
Proof that differentiable functions are continuous proof
Proof that convergent sequences are bounded proof
Proof that convergent sequences are Cauchy proof
Proof that Cauchy sequences are bounded proof
The reals are Archimedean statement
The sequence 1/n tends to zero example
Proof of the term test proof
Proof of the monotone convergence theorem for sequences proof
Proof that a complex number times its conjugate is the square of its modulus proof
Differentiable functions are continuous statement
Limit of a sum of sequences statement
Uniqueness of limits statement
Limits preserve non-strict inequalities statement
Squeeze theorem statement
Convergent sequences are bounded statement
Proof that there is a natural number n with 1/n below any positive epsilon proof
Convergent sequences are Cauchy statement
Cauchy sequences are bounded statement
Monotone convergence theorem for sequences statement
A complex number times its conjugate statement
Proof of the limit of a product of sequences proof
There is a natural number n with 1/n below any positive epsilon statement
Term test statement
Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
The sequence r^n tends to zero for |r| < 1 example
Proof that a decreasing sequence bounded below converges proof
Proof of the sequential characterization of limits proof
Proof that continuity is equivalent to sequential continuity proof
Decreasing sequence bounded below converges statement
Limit of a product of sequences statement
Geometric series example
Proof of the Cauchy–Schwarz inequality proof
A series of nonnegative terms converges iff its partial sums are bounded statement
Continuity is equivalent to sequential continuity statement
Sequential characterization of limits statement
Proof of the root test proof
Proof of the ratio test proof
Proof of the Cauchy–Hadamard theorem proof
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms proof
Proof of the comparison test proof
Cauchy–Schwarz inequality statement
Proof of the chain rule with an auxiliary difference quotient proof
Proof of the product rule proof
Proof of the linearity of differentiation proof
Comparison test statement
Bolzano–Weierstrass theorem statement
Exponentials dominate powers in sequences statement
Chain rule statement
Product rule statement
Linearity of differentiation statement
Ratio test statement
Root test statement
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem proof
Proof of the constant multiple rule proof
Proof of the power rule by induction proof
Boundedness theorem statement
Cauchy–Hadamard theorem statement
Cauchy criterion for sequences statement
Constant multiple rule statement
Power rule statement
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Uniformly Cauchy sequence of functions definition
Proof of the Cauchy criterion for series proof
Extreme value theorem statement
Proof of the formula for the derivative of a polynomial proof
Proof of Fermat's theorem on stationary points proof
Cauchy criterion for series statement
Derivative of a polynomial statement
Proof that absolutely convergent series converge proof
Fermat's theorem on stationary points statement
Absolutely convergent series converge statement
Proof of Rolle's theorem proof
Rolle's theorem statement
Proof of the Weierstrass M-test proof
Proof of the mean value theorem from Rolle's theorem proof
Weierstrass M-test statement
Mean value theorem statement
Proof that power series converge uniformly on smaller intervals inside the radius of convergence proof
Proof that a function with zero derivative on an open interval is constant proof
Proof that uniform convergence of derivatives allows term-by-term differentiation proof
Power series converge uniformly on smaller discs inside the radius of convergence statement
A function with zero derivative is constant statement
Uniform convergence of derivatives allows term-by-term differentiation statement
Proof of term-by-term differentiation proof
Term-by-term differentiation statement
Proof that power series are differentiable inside the radius of convergence proof
Power series are differentiable inside the radius of convergence statement
Proof of the derivative of cosine by term-by-term differentiation proof
Proof of the derivative of sine by term-by-term differentiation proof
Derivative of cosine statement
Derivative of sine statement
Proof of the Pythagorean trigonometric identity by differentiation proof
Pythagorean trigonometric identity statement
Proof of the law of sines from the law of cosines proof

Where this leads

4 pages build on this one.

Tag	`JD65S6DHQA`
URI	`m2://JD65S6DHQA`
Kind	proof
Writing	W0 stub
MSC	51M04
Proves	Law of sines
Uses	Triangle, Angle at a point, Collinear points, Euclidean distance, Length of a vector, Dot product, Law of cosines, Linear independence, Cauchy–Schwarz inequality, Sine and cosine, Pythagorean trigonometric identity
Changed	4 November 2026

Proof of the law of sines from the law of cosines

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