1. Lulama was asked to evaluate the following expression without the use of a calculator: \( \sin \left(180^{\circ}-135^{\circ}\right) \). She answered as follows: \[ \begin{array}{l} \operatorname{Sin}\left(180^{\circ}-135^{\circ}\right)=\sin 180^{\circ}-\sin 135^{\circ} \\ =0-\sin \left(180^{\circ}-45^{\circ}\right) \text { (step 1) } \\ \text { (step2) } \\ = \text { (step 3) } \\ =-\frac{\sin 45^{\circ}}{2} \text { (step 4) }\end{array} \] \( \begin{array}{ll}\text { 1.1. What law or property did Lulama use to write step 1? } & \text { (1) }\end{array} \) 1.2 Do you think Lulama is correct? Motivate your answer?

The law Lulama used in step 1 is the sine of the difference formula, or more appropriately, it can be seen as a misapplication of the sine function properties. Lulama attempted to simplify the expression by relating the sine of angles, but she mistakenly substituted the sine of offsets instead of correctly applying the sine angle subtraction. Lulama's answer is incorrect. The expression \(180^\circ - 135^\circ\) simplifies to \(45^\circ\), and according to the properties of sine, \(\sin(45^\circ) = \frac{\sqrt{2}}{2}\). So the actual simplification should have led her to conclude that \(\sin(45^\circ) = \frac{\sqrt{2}}{2}\) rather than the incorrect result she arrived at.