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$ \tan \theta = \frac{\sin \theta}{\cos \theta} $

Prerequisites

  • Proof by definition of the functions: sine, cosine, and tangent as the ratios of their respective sides in a right triangle.
    • $ \sin \theta = \frac{\mathrm{opposite}}{\mathrm{hypotenuse}} $
    • $ \cos \theta = \frac{\mathrm{adjacent}}{\mathrm{hypotenuse}} $
    • $ \tan \theta = \frac{\mathrm{opposite}}{\mathrm{adjacent}} $

Proof

Definition of Tangent:

$ \tan \theta = \frac{\mathrm{opposite}}{\mathrm{adjacent}} $


$ \tan \theta = \frac{\mathrm{opposite}}{\mathrm{adjacent}} \cdot 1 $

$ \tan \theta = \frac{\mathrm{opposite}}{\mathrm{adjacent}} \cdot \frac{\mathrm{hypotenuse}}{\mathrm{hypotenuse}} $

$ \tan \theta = \frac{ \frac{\mathrm{opposite}}{\mathrm{hypotenuse}}}{\frac{ \mathrm{adjacent}}{\mathrm{hypotenuse}}} $

Definition of Sine and Cosine:

$ \tan \theta = \frac{\sin \theta}{\cos \theta} $

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