MATH 221 Lecture 2

## MATH 221 Lecture 2

Last updated: 10 July 2012

## Angles

Measure angles according to the distance traveled on a circle of radius 1.

Sketch both $x$ and $y$ to get a circle of radius $r$.

The distance $2\pi$ around a circle of radius 1 stretches to $2\pi r$ around a circle of radius $r$. So the circumference of a circle is $2\pi r$ if the circle is radius $r$.

To find the area of a circle first approximate with a polygon inscribed in the circle. The eight triangles form an octagon ${P}_{8}$ in the circle. The area of the octagon ${P}_{8}$ is almost the same as the area of the circle. Unwrap the octagon.

The area of the octagon is the area of the 8 triangles. The area of each triangle is $\frac{1}{2}bh$. So the area of the octagon is $\frac{1}{2}Bh$.

Take the limit as the number of triangles in the interior polygon gets larger and larger (the polygon gets closer and closer to being the circle). Then

Where $B$ is the total base, $h$ is the height of the triangle, $2\pi$ is the length of an unwrapped circle and $r$ is the radius of the circle.

So the area of a circle is $\pi {r}^{2}$ if the circle is radius $r$.

## Trigonometric functions

 ${$ $}$ $}$ $θ$ $cos⁡θ$ $sin⁡θ$ $←$ $\mathrm{sin}\theta$ is the $y$-coordinate of a point at distance $\theta$ on a circle of radius 1 $\mathrm{cos}\theta$ is the $x$-coordinate of a point at distance $\theta$ on a circle of radius 1

$\begin{array}{cccc}\mathrm{tan}\theta =\frac{\mathrm{sin}\theta }{\mathrm{cos}\theta }\text{,}& \mathrm{cot}\theta =\frac{\mathrm{cos}\theta }{\mathrm{sin}\theta }\text{,}& \mathrm{sec}\theta =\frac{1}{\mathrm{cos}\theta }\text{,}& \mathrm{csc}\theta =\frac{1}{\mathrm{sin}\theta }\end{array}$

Since the equation of a circle of radius 1 is ${x}^{2}+{y}^{2}=1$ this forces ${\mathrm{sin}}^{2}\theta +{\mathrm{cos}}^{2}\theta =1$.

The pictures

show that

$sin(-θ)=-sin⁡θandcos(-θ)=cos⁡θ$

Also

show that

$sin⁡0=0andsin⁡π2=1 cos⁡0=1cos⁡π2=0$

Draw the graphs

$y=\mathrm{sin}\theta$

$y=\mathrm{cos}\theta$

by seeing how the $x$ and $y$ coordinates change as you walk around the circle.

Example: Verify $\frac{\mathrm{sec}B}{\mathrm{cos}B}-\frac{\mathrm{tan}B}{\mathrm{cot}B}=1$

$sec⁡Bcos⁡B-tan⁡Bcot⁡B=1cos⁡Bcos⁡B-sin⁡Bcos⁡Bcos⁡Bsin⁡B=1cos2⁡B-sin2⁡Bcos2⁡B =1-sin2⁡Bcos2⁡B=cos2⁡Bcos2⁡B =1$

Example: Verify $\mathrm{cot}\alpha -\mathrm{cot}\beta =\frac{\mathrm{sin}\left(\beta -\alpha \right)}{\mathrm{sin}\alpha \mathrm{sin}\beta }$

$Left hand side=cot⁡α-cot⁡β=cos⁡αsin⁡α-cos⁡βsin⁡β =cos⁡αsin⁡β-cos⁡βsin⁡αsin⁡αsin⁡β Right hand side=sin⁡(β-α)sin⁡αsin⁡β=sin⁡βcos⁡(-α)+cos⁡βsin⁡(-α)sin⁡αsin⁡β =sin⁡βcos⁡α+cos⁡β(-sin⁡α)sin⁡αsin⁡β=sin⁡βcos⁡α-cos⁡βsin⁡αsin⁡αsin⁡β$

So

$\text{Left hand side}=\text{Right hand side}$

Example: Verify $\frac{\mathrm{tan}A-\mathrm{sin}A}{\mathrm{sec}A}=\frac{{\mathrm{sin}}^{3}A}{1+\mathrm{cos}A}$

## References

[Ram] A. Ram, MATH 221 Lecture 2, September 8, 2000, University of Wisconsin.

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