Determine the gradients of the lines with the following angles of inclinat leaving your answers in the simplest surd form: \( \begin{array}{lllll}\text { a) } 45^{\circ} & \text { b) } 30^{\circ} & \text { c) } 150^{\circ} & \text { d) } 0^{\circ} & \text { e) } 270^{\circ}\end{array} \)

To find the gradients (slopes) of the lines at the given angles, we can use the tangent function, where the gradient \( m \) is equal to \( \tan(\theta) \). Let's calculate the gradients for each angle: a) For \( 45^{\circ} \): \[ m = \tan(45^{\circ}) = 1 \] b) For \( 30^{\circ} \): \[ m = \tan(30^{\circ}) = \frac{1}{\sqrt{3}} \] c) For \( 150^{\circ} \): \[ m = \tan(150^{\circ}) = -\frac{1}{\sqrt{3}} \] d) For \( 0^{\circ} \): \[ m = \tan(0^{\circ}) = 0 \] e) For \( 270^{\circ} \): \[ m = \tan(270^{\circ}) \text{ is undefined} \] So, the results in simplest surd form are: - a) \( 1 \) - b) \( \frac{1}{\sqrt{3}} \) - c) \( -\frac{1}{\sqrt{3}} \) - d) \( 0 \) - e) undefined