## 计算机代写|计算机图形学代写computer graphics代考|Units of Angular Measurement

The measurement of angles is at the heart of trigonometry, and today two units of angular measurement are part of modern mathematics: degrees and radians. The degree (or sexagesimal) unit of measure derives from defining one complete rotation as $360^{\circ}$. Each degree divides into $60 \mathrm{~min}$, and each minute divides into $60 \mathrm{~s}$. The number 60 has survived from Mesopotamian days and appears rather incongruous when used alongside today’s decimal system-nevertheless, it is still convenient to work with degrees even though the radian is a natural feature of mathematics.

The radian of angular measure does not depend upon any arbitrary constant, and is often defined as the angle created by a circular arc whose length is equal to the circle’s radius. And because the perimeter of a circle is $2 \pi r, 2 \pi$ rad correspond to one complete rotation. As $360^{\circ}$ corresponds to $2 \pi \mathrm{rad}, 1$ rad equals $180^{\circ} / \pi$, which is approximately $57.3^{\circ}$. The following relationships between radians and degrees are worth remembering:
\begin{aligned} \frac{\pi}{2}[\mathrm{rad}] & \equiv 90^{\circ}, & \pi[\mathrm{rad}] & \equiv 180^{\circ} \ \frac{3 \pi}{2}[\mathrm{rad}] & \equiv 270^{\circ}, & 2 \pi[\mathrm{rad}] & \equiv 360^{\circ} . \end{aligned}
To convert $x^{\circ}$ to radians:
$$\frac{\pi x^{\circ}}{180}\lfloor\mathrm{rad}\rfloor \text {. }$$
To convert $x$ [rad] to degrees:
$$\frac{180 x}{\pi} \text { [degrees]. }$$
For those readers wishing to know the background to radians we need to use power series. We start with the power series for $\mathrm{e}^\theta, \sin \theta$ and $\cos \theta$ :
\begin{aligned} \mathrm{e}^\theta &=1+\frac{\theta^1}{1 !}+\frac{\theta^2}{2 !}+\frac{\theta^3}{3 !}+\frac{\theta^4}{4 !}+\frac{\theta^5}{5 !}+\frac{\theta^6}{6 !}+\frac{\theta^7}{7 !}+\frac{\theta^8}{8 !}+\frac{\theta^9}{9 !}+\cdots \ \sin \theta &=\theta-\frac{\theta^3}{3 !}+\frac{\theta^5}{5 !}-\frac{\theta^7}{7 !}+\frac{\theta^9}{9 !}+\cdots \ \cos \theta &=1-\frac{\theta^2}{2 !}+\frac{\theta^4}{4 !}-\frac{\theta^6}{6 !}+\frac{\theta^8}{8 !}+\cdots . \end{aligned}
Euler proved that these three power series are related, and when $\theta=\pi, \sin \theta=0$, and $\cos \theta=-1$. Figure $4.1$ shows curves of the sine power series for 3, 5, 7 and 9 terms, and when $\theta=2 \pi$, the graph reaches zero.

## 计算机代写|计算机图形学代写computer graphics代考|The Trigonometric Ratios

Ancient civilisations knew that triangles-whatever their size-possessed some inherent properties, especially the ratios of sides and their associated angles. This means that if these ratios are known in advance, problems involving triangles with unknown lengths and angles, can be discovered using these ratios.

Figure $4.2$ shows a point $P$ with coordinates (base, height), on a unit-radius circle rotated through an angle $\theta$. As $P$ is rotated, it moves into the 2nd quadrant, 3 rd quadrant, 4th quadrant and returns back to the first quadrant. During the rotation, the sign of height and base change as follows:
$\begin{array}{ll}\text { 1st quadrant: } & \text { height }(+) \text {, base }(+) \ \text { 2nd quadrant: } & \text { height }(+) \text {, base }(-) \ \text { 3rd quadrant: } & \text { height }(-) \text {, base }(-) \ \text { 4th quadrant: } & \text { height }(-) \text {, base }(+) \text {. }\end{array}$
Figures $4.3$ and $4.4$ plot the changing values of height and base over the four quadrants, respectively. When radius $=1$, the curves vary between 1 and $-1$. In the context of triangles, the sides are labelled as follows:
\begin{aligned} \text { hypotenuse } &=\text { radius } \ \text { opposite } &=\text { height } \ \text { adjacent } &=\text { base } . \end{aligned}
Thus, using the right-angle triangle shown in Fig. 4.5, the trigonometric ratios: sine, cosine and tangent are defined as
$$\sin \theta=\frac{\text { opposite }}{\text { hypotenuse }}, \quad \cos \theta=\frac{\text { adjacent }}{\text { hypotenuse }}, \tan \theta=\frac{\text { opposite }}{\text { adjacent }} .$$

# 计算机图形学代考

## 计算机代写|计算机图形学代写computer graphics代考|Units of Angular Measurement

$$\frac{\pi}{2}[\mathrm{rad}] \equiv 90^{\circ}, \quad \pi[\mathrm{rad}] \equiv 180^{\circ} \frac{3 \pi}{2}[\mathrm{rad}] \quad \equiv 270^{\circ}, 2 \pi[\mathrm{rad}] \quad \equiv 360^{\circ} .$$

$$\frac{\pi x^{\circ}}{180}\lfloor\mathrm{rad}\rfloor .$$

$$\frac{180 x}{\pi} \text { [degrees]. }$$

$$\mathrm{e}^\theta=1+\frac{\theta^1}{1 !}+\frac{\theta^2}{2 !}+\frac{\theta^3}{3 !}+\frac{\theta^4}{4 !}+\frac{\theta^5}{5 !}+\frac{\theta^6}{6 !}+\frac{\theta^7}{7 !}+\frac{\theta^8}{8 !}+\frac{\theta^9}{9 !}+\cdots \sin \theta \quad=\theta-\frac{\theta^3}{3 !}+\frac{\theta^5}{5 !}-\frac{\theta^7}{7 !}+\frac{\theta^9}{9 !}$$

## 计算机代写|计算机图形学代写computer graphics代考|The Trigonometric Ratios

$$\text { hypotenuse }=\text { radius opposite }=\text { height adjacent }=\text { base } \text {. }$$

$$\sin \theta=\frac{\text { opposite }}{\text { hypotenuse }}, \quad \cos \theta=\frac{\text { adjacent }}{\text { hypotenuse }}, \tan \theta=\frac{\text { opposite }}{\text { adjacent }} .$$

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