Solution assignment 04 Unit circle and simple formulas

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Assignment 4

Show using the unit circle:

\sin(-x)=-\sin(x)

\cos(-x)=\cos(x)

\tan(-x)=-\tan(x)

Solution

\sin(-x) has a negative projection on the Y-axis equally large as the positive projection of \sin(x) on the Y-axis.
\cos(-x) has a positive projection on the X-axis equally large and having the same sign as the projection of \cos(x) on the X-axis.
For the third formula we use the following definition of \tan(x) in combination with some formulas concerning \sin(x) and \cos(x):

\tan(x)=\displaystyle\frac{\sin(x)}{\cos(x)}

Thus:

\tan(-x)=\displaystyle\frac{\sin(-x)}{\cos(-x)}=\displaystyle\frac{-\sin(x)}{\cos(x)}=-\tan(x)

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