ข้อสอบ PAT 1 - มีนาคม 2558

$$\begin{gathered} \mathrm{จากโจทย์}\hspace{0.17em}\frac{si{n}^{2}0°+si{n}^{2}10°+si{n}^{2}20°+...+si{n}^{2}170°+si{n}^{2}180°}{co{s}^{2}0°+co{s}^{2}10°+co{s}^{2}20°+...+co{s}^{2}170°+co{s}^{2}180°} \\ \underline{\mathrm{พิจารณา}} si{n}^{2}0°+si{n}^{2}10°+si{n}^{2}20°+...+si{n}^{2}170°+si{n}^{2}180° \\ \mathbf{จากสูตร} \hspace{0.17em}\hspace{0.17em}si{n}^2\left(90+\theta \right)=co{s}^2\theta \mathrm{และ} si{n}^2\theta +co{s}^2\theta =1 \\ \mathrm{จะได้} \hspace{0.17em}\hspace{0.17em}si{n}^{2}0°+si{n}^{2}10°+si{n}^{2}20°+...+si{n}^{2}170°+si{n}^{2}180° \end{gathered}$$

$$\begin{gathered} =si{n}^{2}0°+si{n}^{2}10°+si{n}^{2}20°+... \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+si{n}^{2}90°+si{n}^{2}100°+si{n}^{2}110°+...+si{n}^{2}170°+si{n}^{2}180° \\ =\underset{0}{\underbrace{si{n}^{2}0°}}+si{n}^{2}10°+si{n}^{2}20°+... \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\underset{1}{\underbrace{si{n}^{2}90°}}+\underset{co{s}^{2}10°}{\underbrace{si{n}^{2}100°}}+\underset{co{s}^{2}20}{\underbrace{si{n}^{2}110°}}+...+\underset{co{s}^{2}80}{\underbrace{si{n}^{2}170°}}+\underset{0}{\underbrace{si{n}^{2}180°}} \\ =si{n}^{2}10°+si{n}^{2}20°+si{n}^{2}30°+...+si{n}^{2}80° \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+1+co{s}^{2}10°+co{s}^{2}20°+co{s}^{2}30°+...+co{s}^{2}80°\to \boxed{1} \end{gathered}$$

$$\begin{gathered} \mathbf{โดย} \hspace{0.17em}\hspace{0.17em}{\sin }^{2}10°+{\mathrm{ocs}}^{2}10°=1 \\ {\sin }^{2}20°+{\cos }^{2}20°=1 \\ {\sin }^{2}30°+{\cos }^{2}30°=1 \\ ⋮ ⋮ \\ {\sin }^{2}80°+{\cos }^{2}80°=1 \\ \mathrm{รวม} 8 \mathrm{คู่} \end{gathered}$$
$$\begin{gathered} \mathrm{จาก} \boxed{1} \mathrm{จะได้} \\ {\sin }^{2}0°+{\sin }^{2}10°+{\sin }^{2}20°+...+{\sin }^{2}170°+{\sin }^{2}180°=8\left(1\right)+1 \\ \hspace{0.17em} \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=9 \end{gathered}$$

$$\begin{gathered} \underline{\mathrm{พิจารณา}} co{s}^{2}0°+co{s}^{2}10°+co{s}^{2}20°+...+co{s}^{2}170°+co{s}^{2}180° \\ \mathrm{จากสูตร} \hspace{0.17em}\hspace{0.17em}co{s}^2\left(90+\theta \right)=si{n}^2\theta \mathrm{และ} si{n}^2\theta +co{s}^2\theta =1 \\ \mathrm{จะได้} co{s}^{2}0°+co{s}^{2}10°+co{s}^{2}20°+...+co{s}^{2}170°+co{s}^{2}180° \end{gathered}$$

$$\begin{gathered} =co{s}^{2}0°+co{s}^{2}10°+co{s}^{2}20°+... \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+co{s}^{2}90°+co{s}^{2}100°+co{s}^{2}110°+...+co{s}^{2}170°+co{s}^{2}180° \\ =\underset{1}{\underbrace{co{s}^{2}0°}}+co{s}^{2}10°+co{s}^{2}20°+... \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\underset{0}{\underbrace{co{s}^{2}90°}}+\underset{si{n}^{2}10}{\underbrace{co{s}^{2}100°}}+\underset{si{n}^{2}20}{\underbrace{co{s}^{2}110°}}+...+\underset{si{n}^{2}80}{\underbrace{co{s}^{2}170°}}+\underset{1}{\underbrace{co{s}^{2}180°}} \\ =1+co{s}^{2}10°+co{s}^{2}20°+... \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+0+si{n}^{2}10°+si{n}^{2}20°+...+si{n}^{2}80°+1\to \boxed{2} \end{gathered}$$

$$\begin{gathered} \mathbf{โดย} \hspace{0.17em}{\cos }^{2}10°+{\sin }^{2}10°=1 \\ {\cos }^{2}20°+{\sin }^{2}20°=1 \\ ⋮ ⋮ \\ {\cos }^{2}80°+{\sin }^{2}80°=1 \\ \mathrm{รวม} 8 \mathrm{คู่} \end{gathered}$$

$$\begin{gathered} \mathrm{จาก} \boxed{1} \mathrm{จะได้} \\ {\cos }^{2}0°+{\cos }^{2}10°+{\cos }^{2}20°+...+{\cos }^{2}170°+{\cos }^{2}180°=1+8\left(1\right)+1 \\ =10 \end{gathered}$$

$$\begin{gathered} \mathrm{ดังนั้น} \frac{{\sin }^{2}0°+{\sin }^{2}10°+{\sin }^{2}20°+...+{\sin }^{2}170°+{\sin }^{2}180°}{{\cos }^{2}0°+{\cos }^{2}10°+{\cos }^{2}20°+...+{\cos }^{2}170°+{\cos }^{2}180°}=\frac{\mathrm{a}}{\mathrm{b}} \\ \frac{9}{10}=\frac{\mathrm{a}}{\mathrm{b}} \\ \mathrm{จากโจทย์}\hspace{0.17em} \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{ห}.\mathrm{ร}.\mathrm{ม}. \mathrm{ของ} \mathrm{a} \mathrm{และ} \mathrm{b} \mathrm{เท่ากับ} 1 \\ \mathrm{จะได้} \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{a}=9 , \mathrm{b}=10 \\ \mathrm{ดังนั้น} \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathrm{a}}^2+{\mathrm{b}}^2=9^2+{10}^2 \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=81+100 \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=181 \end{gathered}$$