4. If \( \tan 22^{\circ}=t \) write the following in terms of \( t \). \( \begin{array}{ll}\text { (a) } \tan 202^{\circ} & \text { (b) } \tan 338^{\circ}\end{array} \)

To express \( \tan 202^{\circ} \) and \( \tan 338^{\circ} \) in terms of \( t = \tan 22^{\circ} \), we'll use the periodic and symmetry properties of the tangent function. ### (a) \( \tan 202^{\circ} \) **Step 1: Express 202° in terms of 180° and 22°** \[ 202^{\circ} = 180^{\circ} + 22^{\circ} \] **Step 2: Use the periodicity of tangent** The tangent function has a period of 180°, meaning: \[ \tan(\theta + 180^{\circ}) = \tan \theta \] **Applying this:** \[ \tan 202^{\circ} = \tan(180^{\circ} + 22^{\circ}) = \tan 22^{\circ} = t \] ### (b) \( \tan 338^{\circ} \) **Step 1: Express 338° in terms of 360° and 22°** \[ 338^{\circ} = 360^{\circ} - 22^{\circ} \] **Step 2: Use the symmetry of tangent** The tangent function has the property: \[ \tan(360^{\circ} - \theta) = -\tan \theta \] **Applying this:** \[ \tan 338^{\circ} = \tan(360^{\circ} - 22^{\circ}) = -\tan 22^{\circ} = -t \] ### **Final Answers:** - **(a)** \( \tan 202^{\circ} = t \) - **(b)** \( \tan 338^{\circ} = -t \)