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What do sin and tan mean respectively in mathematics?
Sin is a symbol of sine; Tan is a symbol of tangency.

Sine, a mathematical term, in a right triangle, the ratio of the opposite side to the hypotenuse of any acute angle ∠A is called sine of ∠A, which is abbreviated from the English word sine, that is, the opposite side/hypotenuse of Sina = ∠ A.

In Rt△ABC (right triangle), ∠C = 90, AB is the opposite side C of ∠ C, BC is the opposite side A of ∠A, AC is the opposite side B of ∠B, and the tangent function is tanB=b/a, that is, tanB=AC/BC.

Extended data:

Common sum angle formula

sin(α+β)=sinαcosβ+ sinβcosα

sin(α-β)=sinαcosβ-sinB*cosα

cos(α+β)=cosαcosβ-sinαsinβ

cos(α-β)=cosαcosβ+sinαsinβ

tan(α+β)=(tanα+tanβ)/( 1-tanαtanβ)

tan(α-β)=(tanα-tanβ)/( 1+tanαtanβ)

Double angle formula

sin2α=2sinαcosα

tan2α=2tanα/( 1-tan^2(α))

cos2α=cos^2(α)-sin^2(α)=2cos^2(α)- 1= 1-2sin^2(α)?