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

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

The functions $\sin \theta, \cos \theta, \tan \theta, \csc \theta, \sec \theta$ and $\cot \theta$ provide different ratios for the angle $\theta$, and the inverse trigonometric functions convert a ratio back into an angle. Theses aree arcsinh, arccoos, arctann, arccscc, arcsēc ānd arccot, and aree sometimes writteñ as $\sin ^{-1}, \cos ^{-1}, \tan ^{-1}, \csc ^{-1}, \sec ^{-1}$ and $\cot ^{-1}$. For example, $\sin 30^{\circ}=0.5$, therefore, $\arcsin 0.5=30^{\circ}$. Consequently, the domain for arcsin is the range for sin:
$$[-1,1]$$
and the range for $\arcsin$ is the domain for sin:
$$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$
as shown in Fig. 4.8. Similarly, the domain for arccos is the range for cos:
$$[-1,1]$$
and the range for arccos is the domain for $\cos$ :
$$\lfloor 0, \pi\rfloor$$
as shown in Fig. 4.9.

The domain for arctan is the range for tan:
$$]-\infty, \infty[$$
and the range for arctan is the domain for tan:
$$]-\frac{\pi}{2}, \frac{\pi}{2}[$$
as shown in Fig. $4.10$.
Various programming languages include the atan2 function, which is an arctan function with two arguments: $\operatorname{atan} 2(y, x)$. The signs of $x$ and $y$ provide sufficient information to locate the quadrant containing the angle, and gives the atan 2 function a range of $[0,2 \pi]$.

# 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|The Trigonometric Ratios

1st quadrant: height $(+)$, base (+) 2nd quadrant: height $(+)$, base (-) 3rd quadrant: 数字 $4.3$ 和 $4.4$ 分别在四个象限上绘制高度和底部的变化值。当半径 $=1$ ，曲线在 1 和 $-1$. 在三角形的上 下文中，边标记如下:
$$\text { hypotenuse }=\text { radius opposite }=\text { height adjacent }=\text { base } \text {. }$$

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

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

$$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$

$$[-1,1]$$
$\arccos$ 的范围是 $\cos$ :
$$\lfloor 0, \pi\rfloor$$

$\arctan$ 的域是 $\tan$ 的范围:
$$]-\infty, \infty[$$
$\arctan$ 的范围是 $\tan$ 的定义域:
$$]-\frac{\pi}{2}, \frac{\pi}{2}[$$

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