trigonometric ratios: why the tangent of a 45 angle is 1

[minela22]

Please help me solve this problem urgent!
Explain why the tangent of a 45 angle is 1

 

[o_O]

Re: trigonometric ratios

Do you know the identity:
$\displaystyle tanx = \frac{sinx}{cosx}$ ?

If so, then what is sin45 and cos45 equal to?

If not, I'll prove it to you. Imagine a triangle $\displaystyle \triangle ABC$ with sides a, b, c, $\displaystyle \angle ACB$ is a right angle, and $\displaystyle x = \angle ABC$

  A
   *
   **
   *  *
 b  *    *   c
   *      *
   *        *
   * * * * * * B
  C     a

From that, we have by definition:
$\displaystyle sinx = \frac{b}{c} \quad cosx = \frac{a}{c} \quad tanx = \frac{b}{a}$

That means:
$\displaystyle \frac{sinx}{cosx} = \frac{\:\frac{b}{c}\:}{\:\frac{a}{c}\:} = \frac{b}{c} \cdot \frac{c}{a} = \frac{bc}{ac} = \frac{b}{a} = tanx$

$\displaystyle \therefore tanx = \frac{sinx}{cosx}$