The sine and cosine are related to the definitions because of the identity [tex]\cos^{2}\theta + \sin^{2}\theta = 1[/tex].
How cosine and sine are related in the unit circle
The unit circle is a particular case of the Pythagorean theorem, in which hypotenuse ([tex]r[/tex]) has a measure of a unit. That is:
[tex]x^{2}+y^{2} = r^{2}[/tex] (1)
Where:
- [tex]x[/tex] - Horizontal leg
- [tex]y[/tex] - Vertical leg
If we divide the entire expression by [tex]r^{2}[/tex], then we have the following identity:
[tex]\left(\frac{x}{r} \right)^{2}+\left(\frac{y}{r} \right)^{2} = 1[/tex]
And by trigonometric relations, we have the following formula:
[tex]\cos^{2}\theta + \sin^{2}\theta = 1[/tex] (2)
The sine and cosine are related to the definitions because of the identity [tex]\cos^{2}\theta + \sin^{2}\theta = 1[/tex]. [tex]\blacksquare[/tex]
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