What are what are sin tan and cos and how do they relate to the Pythagorean theorem?

Sunday, February 22, 2009

What are what are sin tan and cos and how do they relate to the Pythagorean theorem?

sin, cos, and tan can be considered as trigonometric functions or as ratios of sides of a right triangle. We will use the second interpretation.

In a right triangle, consider one of the acute angles; let it be A. The other acute angle will be B and the right angle C with the sides opposite the angles labelled a,b, and c respectively.

The sin (sine) is defined as the ratio of the side opposite the angle to the hypotenuse; so $\displaystyle{\sin{{A}}}=\frac{{a}}{{c}}$

The cos (cosine) is defined as the ratio of the side adjacent the angle to the hypotenuse; so $\displaystyle{\cos{{A}}}=\frac{{b}}{{c}}$

The tan (tangent) is defined as the ratio of the leg opposite the angle to the leg adjacent the angle; so $\displaystyle{\tan{{A}}}=\frac{{a}}{{b}}$

The Pythagorean theorem says that for this right triangle $\displaystyle{{a}}^{{2}}+{{b}}^{{2}}={{c}}^{{2}}$ .

If we take $\displaystyle{{\sin}}^{{2}}{A}+{{\cos}}^{{2}}{A}$ we get $\displaystyle{{\left(\frac{{a}}{{c}}\right)}}^{{2}}+{{\left(\frac{{b}}{{c}}\right)}}^{{2}}$ or $\displaystyle\frac{{{{a}}^{{2}}+{{b}}^{{2}}}}{{{c}}^{{2}}}$ ; using the Pythagorean theorem we can rewrite the numerator as $\displaystyle{{c}}^{{2}}$ so that:

$\displaystyle{{\sin}}^{{2}}{A}+{{\cos}}^{{2}}{A}={1}$

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Sunday, February 22, 2009