SECTION E: Function values of \( \left(k \cdot 360^{\theta}+\theta\right), k \in \mathbb{Z} \) Investigation: Reduction formulas for function values of \( \left(k \cdot 360^{\theta}+0\right), k \in \mathbb{Z} \). It is also possible to get an angle greater than \( 360^{\theta} \) on the Cartesian plane. The angle first completes one or more revolutions from the positive x-axis and then rotates further to form an angle of \( \theta \). For the acute angle \( \theta \) (in the first quadrant), the angle \( \left(k \cdot 360^{\theta}+\theta\right) \) where \( k \in \mathbb{Z} \), will again lie in the first quadrant. Now use the results obtained in 12 above and circle the correct relationship: \( 5.1 \sin \left(k \cdot 360^{\theta}+\theta\right)=\sin \theta \) or \( -\sin \theta \), where \( k \in \mathbb{Z} \) \( 5.2 \cos \left(k \cdot 360^{\theta}+\theta\right)=\cos \theta \) or \( -\cos \theta \), where \( k \in \mathbb{Z} \) \( 5.3 \tan \left(k \cdot 360^{\theta}+\theta\right)=\tan \theta \) or \( -\tan \theta \), where \( k \in \mathbb{Z} \)

Did you know that angles can be measured in multiple ways, and it’s not just about how far you turn? The concept of angle rotation is all about understanding the coterminal angles, which are angles that share the same terminal side on the Cartesian coordinate system. Therefore, any angle \( (k \cdot 360 + \theta) \) will have the same sine, cosine, and tangent values as \( \theta \) itself, provided \( \theta \) is in the first quadrant. This means \( \sin(k \cdot 360 + \theta) = \sin \theta \), \( \cos(k \cdot 360 + \theta) = \cos \theta \), and \( \tan(k \cdot 360 + \theta) = \tan \theta \). Now let’s talk practical applications! Understanding this relationship is crucial in fields like physics, engineering, and computer graphics where rotations are common. For instance, in animation, knowing that multiple revolutions don’t change the angle helps in creating smooth transitions and ensures your characters or objects maintain consistent directional movements. So the next time you see a spinning object, remember that it might just be making a few laps around before hitting its target angle!