The Cosine Rule

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From Pythagoras’ theorem we know that

$${b^2 = h_c^2 + (c - d)^2}$$

Expanding the bracket:

$${b^2 = {\color{orange}{h_c^2}} + c^2 - 2cd + d^2}$$

Also,

$${a^2 = h_c^2 + d^2}$$

$${{\color{orange}{h_c^2 = a^2 - d^2}}}$$

Substituting, we get

$${b^2 = {\color{orange}{a^2 - d^2}} + c^2 - 2cd + d^2}$$

$${b^2 = a^2 + c^2 - 2c{\color{yellow}{d}}}$$

From the trigonometric ratios, we also know that

$${\cos{\beta} = \frac{d}{a} }$$

$${\color{yellow}{d = a \cdot \cos{\beta} } }$$

Substituting for d:

$${b^2 = a^2 + c^2 - 2{\color{yellow}{a }}c\cdot{\color{yellow}{\cos{\beta}}}}$$

the cosine rule - der Kosinussatz

$${b^2 = a^2 + c^2 - 2ac\cdot\cos{\beta}}$$

If looking for the angle given the three sides.

$${\cos{\beta} = \frac{a^2 + c^2 - b^2} {2ac}}$$

Remember that the square of the side opposite the given angle is subtracted from the sum of squares of the two sides enclosing the angle.

$${\beta = \cos^{-1}\left(\frac{a^2 + c^2 - b^2} {2ac}\right)}$$

or

$${\beta = \arccos\left(\frac{a^2 + c^2 - b^2} {2ac}\right)}$$

applications

CIMT - exercises (same as on sine rule page)


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