ข้อสอบ PAT 1 - พฤศจิกายน 2557

$$\begin{gathered} \mathbf{สูตรผลบวก}\mathbf{ }\mathbf{ผลต่างตรีโกณ} \\ \sin \left(\mathrm{A}+\mathrm{B}\right)+\sin \left(\mathrm{A}-\mathrm{B}\right)=2\sin \mathrm{A} \sin \mathrm{B} \\ \sin \left(\mathrm{A}+\mathrm{B}\right)-\sin \left(\mathrm{A}-\mathrm{B}\right)=2\cos \mathrm{A} \cos \mathrm{B} \\ \cos \left(\mathrm{A}+\mathrm{B}\right)+\cos \left(\mathrm{A}-\mathrm{B}\right)=2\cos \mathrm{A} \cos \mathrm{B} \\ \cos \left(\mathrm{A}+\mathrm{B}\right)-\cos \left(\mathrm{A}-\mathrm{B}\right)=-2\sin \mathrm{A} \sin \mathrm{B} \end{gathered}$$

$$\begin{gathered} \mathrm{จากโจทย์} \frac{\sin 25°\sin 85°\sin 35°}{\sin 75°} ; \mathrm{คูณ} -2 \mathrm{บน} \mathrm{ล่าง} \\ =\frac{\left(-2\sin 25°\sin 85°\right)\sin 35°}{-2\sin 75°} \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathbf{\hspace{0.17em}}\mathbf{โดย} -2\sin \mathrm{A} \sin \mathrm{B}=\cos \left(\mathrm{A}+\mathrm{B}\right)-\cos \left(\mathrm{A}-\mathrm{B}\right) \end{gathered}$$

$$\begin{gathered} =\frac{\left[\cos \left(25+85\right)-\cos \left(25-85\right)\right]\sin 35°}{-2\sin 75°} \\ =\frac{\left[\cos 110°-\cos \left(-60\right)°\right]\sin 35°}{-2\sin 75°} \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathbf{โดย} \cos \left(-\mathrm{\theta }\right)=\mathrm{cos\theta } ; \\ =\frac{\left[\cos 110°-{\displaystyle \frac{1}{2}}\right]\sin 35°}{-2\sin 75°} \\ =\frac{\cos 110°\sin 35°-{\displaystyle \frac{1}{2}}\sin 35°}{-2\sin 75°} \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{คูณ} 2 \mathrm{ทั้งบน} \mathrm{ล่าง} ; \end{gathered}$$

$$\begin{gathered} =\frac{2\cos 110°\sin 35°-\sin 35°}{-4\sin 75°} \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathbf{โดย} 2\cos \mathrm{A} \mathrm{sinB}=\sin \left(\mathrm{A}+\mathrm{B}\right)-\sin \left(\mathrm{A}-\mathrm{B}\right) \\ =\frac{\sin \left(110+35\right)-\sin \left(110-35\right)-\sin 35°}{-4\sin 75°} \\ =\frac{\sin 145°-\sin 75°-\sin 35°}{-4\sin 75°} \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathbf{\hspace{0.17em}}\mathbf{โดย} \sin 145°=\sin 35° ; \\ =\frac{\cancel{\sin 35°}-\sin 75°-\cancel{\sin 35°}}{-4\sin 75°} \\ =\frac{-\cancel{\sin 75°}}{-4\cancel{\sin 75°}} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{=}\frac{1}{4} \end{gathered}$$

$$\begin{gathered} \underline{\mathrm{พิจารณาตัวเลือก}} \\ 1. \tan 15°=\frac{\sqrt{3}-1}{\sqrt{3+1}} \\ 2. \sin 15°\sin 75°=\frac{-2\sin 15°\sin 75°}{-2} \\ \mathrm{คูณ} -2 \mathrm{บน} \mathrm{ล่าง} \\ =\frac{\cos (15+75)-\cos (15-75)}{-2} \\ =\frac{\cos 90°-\cos (-60)°}{-2} \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{โดย} \cos (-\mathrm{\theta })=\mathrm{cos\theta } \\ =\frac{0-{\displaystyle \frac{1}{2}}}{-2} \\ =\frac{1}{4} \end{gathered}$$

$$\begin{gathered} 3. \cos 20°\cos 40°\cos 80°=\frac{\left(2\cos 20°\cos 40°\right)\cos 80°}{2} \\ \mathrm{คูณ} 2 \mathrm{บน} \mathrm{ล่าง} ; =\frac{\left[\cos \left(20+40\right)+\cos \left(20-40\right)\right]\cos 80°}{2} \\ =\frac{\left[\cos 60°+\cos \left(-20\right)°\right]\cos 80°}{2} \end{gathered}$$

$$\begin{gathered} \cos (-\mathrm{\theta })=\mathrm{cos\theta } ; =\frac{[{\displaystyle \frac{1}{2}}+\cos 20°]\cos 80°}{2} \\ =\frac{{\displaystyle \frac{1}{2}\cos 80°}+\cos 20°\cos 80°}{2} \\ =\frac{{\displaystyle \cos 80°+\left(2\cos 20°\cos 80°\right)}}{4} \\ \mathrm{โดย} 2\cos \mathrm{A} \cos \mathrm{B}=\cos \left(\mathrm{A}+\mathrm{B}\right)+\cos \left(\mathrm{A}-\mathrm{B}\right) ; \\ \mathrm{จะได้} =\frac{{\displaystyle \cos 80°+\cos 100°+\cos \left(-60\right)°}}{4} \\ \cos 100°=-\cos 80° ; =\frac{{\displaystyle \cancel{\cos 80°}+\left(-\cancel{\cos 80°}\right)+\cos \left(-60\right)°}}{4} \end{gathered}$$

$$\begin{gathered} =\frac{{\displaystyle \cos \left(-60\right)°}}{4}=\frac{{\displaystyle \cos 60°}}{4} \\ \cos 60°=\frac{1}{2} ; =\frac{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{4} \\ =\frac{{\displaystyle 1}}{8} \\ 4. \mathrm{sc}420°=\sec \left(360+60\right)° \\ =\sec 60° \\ =2 \end{gathered}$$