15 4-4-2 الزاويتان 180° − θ و 180° + θ
- \[\begin{gathered}\sin (180{}^{\circ} - \theta ) = \sin \theta \end{gathered}\] و \[\begin{gathered}\cos (180{}^{\circ} - \theta ) = -\cos \theta \end{gathered}\] و \[\begin{gathered}\tan (180{}^{\circ} - \theta ) = -\tan \theta \end{gathered}\].
- وكذلك \[\begin{gathered}\sin (180{}^{\circ} + \theta ) = -\sin \theta \end{gathered}\] و \[\begin{gathered}\cos (180{}^{\circ} + \theta ) = -\cos \theta \end{gathered}\] و \[\begin{gathered}\tan (180{}^{\circ} + \theta ) = \tan \theta \end{gathered}\].
\[\begin{gathered}\sin (180{}^{\circ} - \theta ) = \sin \theta \end{gathered}\]
\[\begin{gathered}\cos (180{}^{\circ} - \theta ) = -\cos \theta \end{gathered}\]
\[\begin{gathered}\tan (180{}^{\circ} - \theta ) = -\tan \theta \end{gathered}\]
\[\begin{gathered}\sin (180{}^{\circ} + \theta ) = -\sin \theta \end{gathered}\]
\[\begin{gathered}\cos (180{}^{\circ} + \theta ) = -\cos \theta \end{gathered}\]
\[\begin{gathered}\tan (180{}^{\circ} + \theta ) = \tan \theta \end{gathered}\]
- نفرض θ حادة، فالزاوية 180° − θ في الربع الثاني والزاوية 180° + θ في الربع الثالث.
- نحتفظ باسم الدالة (لا تتحول) ونحدد الإشارة من الربع.
لاحظ: عند الأعداد 180° و 360° (مضاعفات 180°) لا يتغير اسم الدالة.
مثال 1: أوجد قيمة sin 150° و cos 150°.
الحل:
\[\begin{gathered}150{}^{\circ} = 180{}^{\circ} - 30{}^{\circ}\end{gathered}\]
\[\begin{gathered}\sin 150{}^{\circ} = \sin 30{}^{\circ} = \dfrac{1}{2}\end{gathered}\]
\[\begin{gathered}\cos 150{}^{\circ} = -\cos 30{}^{\circ} = -\dfrac{\sqrt{3}}{2}\end{gathered}\]
مثال 2: أوجد قيمة cos 210° و tan 210°.
الحل:
\[\begin{gathered}210{}^{\circ} = 180{}^{\circ} + 30{}^{\circ}\end{gathered}\]
\[\begin{gathered}\cos 210{}^{\circ} = -\cos 30{}^{\circ} = -\dfrac{\sqrt{3}}{2}\end{gathered}\]
\[\begin{gathered}\tan 210{}^{\circ} = \tan 30{}^{\circ} = \dfrac{1}{\sqrt{3}}\end{gathered}\]
مثال 3: أوجد قيمة sin 225°.
الحل:
\[\begin{gathered}225{}^{\circ} = 180{}^{\circ} + 45{}^{\circ}\end{gathered}\]
\[\begin{gathered}\sin 225{}^{\circ} = -\sin 45{}^{\circ} = -\dfrac{\sqrt{2}}{2}\end{gathered}\]
مثال 4: أوجد قيمة tan 135°.
الحل:
\[\begin{gathered}135{}^{\circ} = 180{}^{\circ} - 45{}^{\circ}\end{gathered}\]
\[\begin{gathered}\tan 135{}^{\circ} = -\tan 45{}^{\circ} = -1\end{gathered}\]
تذكر: الأخطاء الشائعة: قلب اسم الدالة هنا (لا يتغير عند 180°)؛ إهمال إشارة الربع.