Tangent theorem

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In the trigonometry , the law of tangents (also tangent set and control of Napier ) a relationship between the three sides of a plane triangle and the tangent of half the sum and half the difference of two angles of the triangle forth.

For the three sides a , b and c of a triangle as well as for the angles α, β and γ lying opposite these sides:

Because of this, this formula can also be written as

Analogous formulas for and are obtained through cyclic exchange :

Because of this , one of these formulas remains valid if both the sides and the associated angles are swapped, for example:

Proof with the law of sines and identities of the trigonometric functions

According to the law of sines , it follows that:

after inserting the identities

such as

,

which can be derived from the addition theorems , the desired formula is obtained by division.

Proof with Mollweid's formulas

With the sum of the angles in the triangle and transition to the complementary angle :

(1)

From Mollweid's formulas it follows with (1):

qed

See also

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