Law of Cosines

The first equation will be proven for obtuse angles. The remaining equations can be proven similarly.

a^{2} = b^{2} + c^{2} - 2 b c cos A

Consider

△ A B C

with side lengths of

a,

b,

and

c,

respectively opposite the angles with measures

A,

B,

and

C,

such that

m ∠ A

is greater than

9 0^{\circ} .

The altitude of the triangle is the perpendicular segment from $B$ to the extension of the base $\overline{A C} .$ Let $D$ be the endpoint of this segment and $x$ be the distance from $D$ to $A .$

From the definition of an altitude, it follows that

△ B D A

and

△ B D C

are right triangles. Two equations can be obtained by applying the Pythagorean Theorem to both triangles.

Equation I : Equation II : a^{2} = h^{2} + (b + x)^{2} c^{2} = h^{2} + x^{2}

Expand the binomial in Equation I.

a^{2} = h^{2} + (b + x)^{2}

ExpandPosPerfectSquare

$(a + b)^{2} = a^{2} + 2 a b + b^{2}$

a^{2} = h^{2} + b^{2} + 2 b x + x^{2}

CommutativePropAdd

Commutative Property of Addition

a^{2} = h^{2} + x^{2} + b^{2} + 2 b x

From Equation II,

h^{2} + x^{2}

is equal to

c^{2} .

Therefore, the expanded form of Equation I can be rewritten by using the Substitution Property of Equality.

a^{2} = h^{2} + x^{2} + b^{2} + 2 b x

Substitute

$h^{2} + x^{2} = c^{2}$

a^{2} = c^{2} + b^{2} + 2 b x

Note that

A

and

18 0^{\circ} - A

are supplementary angles. Using the cosine ratio of

18 0^{\circ} - A

then gives an expression for the

x -

term.

△ B D A,

the cosine of

18 0^{\circ} - A

is the ratio of

x

c .

cos (18 0^{\circ} - A) = \frac{x}{c} ⇕ x = c cos (18 0^{\circ} - A)

By the Sine and Cosine of Supplementary Values Angles,

cos (18 0^{\circ} - A)

and

cos A

have opposite values.

x = c cos (18 0^{\circ} - A) ⇕ x = - c cos A

Finally, by substituting

- c c o s A

for

x

into

a^{2} = b^{2} + c^{2} + 2 b x,

the Law of Cosines is obtained.

a^{2} = b^{2} + c^{2} + 2 b x

Substitute

$x = - c c o s A$

a^{2} = b^{2} + c^{2} + 2 b (- c c o s A)

MultPosNeg

$a (- b) = - a \cdot b$

a^{2} = b^{2} + c^{2} - 2 b c cos A

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Law of Cosines

Proof

Proof

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