They are different trigonometric functions!
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Tangent = sine/cosine provided that cosine is non-zero. When cosine is 0, then tangent is undefined.
Yes, sine, cosine, tangent definitions are based on right triangles
No, it does not.
Trigonometry
It depends on what information you already have. For example, if you know the length of two sides of a triangle, you can easily find the tangent. Or, if you know the length of two angles and a side, you can find the other sides as well, using the tangent, cosine, and sine as needed.