Prove why Tan equals sin/cos?

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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In a right triangle ABC, whose right angle is A, the tangent ratio of the angle B is the opposite leg over the adjacent leg.

tan B = opposite/adjacent

We'll express the sine and cosine functions of the angle B:

sin B = opposite/hypotenuse => opposite = hypotenuse*sin B

cos B = adjacent/hypotenuse => adjacent = hypotenuse*cos B

We'll re-write the tangent ratio:

tan B = opposite/adjacent <= > tan B = hypotenuse*sin B/hypotenuse*cos B

We'll reduce hypotenuse:

tan B = sin B/cos B

Therefore, in a right angle triangle, the tangent ratio of an angle, except the right angle, is: tan `alpha` = sin `alpha` / cos `alpha` .