If \( \sin 43^{\circ}=\frac{1}{a} \), express the following in terms of \( a \) : \( 3.1 \sin 223^{\circ} \) \( 3.2 \cos 317^{\circ} \) \( 3.3 \tan \left(-137^{\circ}\right) \) \( 3.4 \cos 493^{\circ} \)

1. We are given \[ \sin 43^\circ=\frac{1}{a}. \] 2. For each part, we express the trigonometric function in terms of \(\sin 43^\circ\) (or its related cosine/tangent). --- **3.1 \(\sin 223^\circ\)** Note that \[ 223^\circ=180^\circ+43^\circ, \] and using the identity \[ \sin(180^\circ+\theta)=-\sin \theta, \] we have \[ \sin 223^\circ=-\sin 43^\circ=-\frac{1}{a}. \] --- **3.2 \(\cos 317^\circ\)** Observe that \[ 317^\circ=360^\circ-43^\circ, \] and the cosine identity gives \[ \cos(360^\circ-\theta)=\cos \theta, \] so \[ \cos 317^\circ=\cos 43^\circ. \] Using the Pythagorean identity, we can express \(\cos 43^\circ\) in terms of \(\sin 43^\circ\): \[ \cos 43^\circ=\sqrt{1-\sin^2 43^\circ}=\sqrt{1-\frac{1}{a^2}}. \] Thus, \[ \cos 317^\circ=\sqrt{1-\frac{1}{a^2}}. \] --- **3.3 \(\tan(-137^\circ)\)** We use the periodicity of the tangent function. Notice that \[ -137^\circ+180^\circ=43^\circ, \] and since \[ \tan(\theta-180^\circ)=\tan \theta, \] we have \[ \tan(-137^\circ)=\tan 43^\circ. \] Express \(\tan 43^\circ\) in terms of sine and cosine: \[ \tan 43^\circ=\frac{\sin 43^\circ}{\cos 43^\circ}=\frac{\frac{1}{a}}{\sqrt{1-\frac{1}{a^2}}}. \] Simplify the expression: \[ \sqrt{1-\frac{1}{a^2}}=\sqrt{\frac{a^2-1}{a^2}}=\frac{\sqrt{a^2-1}}{a}, \] so, \[ \tan 43^\circ=\frac{\frac{1}{a}}{\frac{\sqrt{a^2-1}}{a}}=\frac{1}{\sqrt{a^2-1}}. \] Thus, \[ \tan(-137^\circ)=\frac{1}{\sqrt{a^2-1}}. \] --- **3.4 \(\cos 493^\circ\)** First, subtract \(360^\circ\) to find a coterminal angle: \[ 493^\circ-360^\circ=133^\circ, \] so \[ \cos 493^\circ=\cos 133^\circ. \] Recognize that \[ 133^\circ=90^\circ+43^\circ, \] and using the identity \[ \cos(90^\circ+\theta)=-\sin \theta, \] we get \[ \cos 133^\circ=-\sin 43^\circ=-\frac{1}{a}. \] Thus, \[ \cos 493^\circ=-\frac{1}{a}. \] --- Final answers: \[ \begin{aligned} 3.1\quad \sin 223^\circ&=-\frac{1}{a},\\[1mm] 3.2\quad \cos 317^\circ&=\sqrt{1-\frac{1}{a^2}},\\[1mm] 3.3\quad \tan(-137^\circ)&=\frac{1}{\sqrt{a^2-1}},\\[1mm] 3.4\quad \cos 493^\circ&=-\frac{1}{a}. \end{aligned} \]