\begin{tabular}{|l|} d) $$ 490^{\circ} $$ \ \#3.) List all angles coterminal to $$ 500^{\circ} $$ where $$ -720^{\circ}

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\begin{tabular}{|l|} d) $$ 490^{\circ} $$ \ \#3.) List all angles coterminal to $$ 500^{\circ} $$ where $$ -720^{\circ}< $$ \ $$ 	heta<360^{\circ} $$\end{tabular}

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We wish to find all angles that are coterminal to $$500^\circ$$ and lie in the interval $$ -720^\circ < \theta < 360^\circ. $$ Two angles are coterminal if they differ by an integer multiple of $$360^\circ$$. That is, an angle $$\theta$$ is coterminal with $$500^\circ$$ if $$ \theta = 500^\circ + 360^\circ k, $$ where $$k$$ is an integer. **Step 1. Set Up the Inequality** We substitute the expression for $$\theta$$ into the given range: $$ -720^\circ < 500^\circ + 360^\circ k < 360^\circ. $$ **Step 2. Solve for $$k$$ in the Lower Bound** Starting with $$ 500^\circ + 360^\circ k > -720^\circ, $$ subtract $$500^\circ$$ from both sides: $$ 360^\circ k > -720^\circ - 500^\circ = -1220^\circ. $$ Now divide by $$360^\circ$$: $$ k > \frac{-1220}{360} \approx -3.39. $$ Since $$k$$ is an integer, the smallest possible integer satisfying this inequality is $$ k \geq -3. $$ **Step 3. Solve for $$k$$ in the Upper Bound** Now consider $$ 500^\circ + 360^\circ k < 360^\circ. $$ Subtract $$500^\circ$$ from both sides: $$ 360^\circ k < 360^\circ - 500^\circ = -140^\circ. $$ Divide by $$360^\circ$$: $$ k < \frac{-140}{360} \approx -0.3889. $$ Thus, $$k$$ must be less than approximately $$-0.3889$$, so the greatest integer allowed is $$ k \leq -1. $$ **Step 4. List the Possible Values of $$k$$** The integer values of $$k$$ that satisfy both conditions $$-3 \leq k \leq -1$$ are: $$ k = -3,\quad -2,\quad -1. $$ **Step 5. Compute the Corresponding Angles** For $$k = -3$$: $$ \theta = 500^\circ + 360^\circ(-3) = 500^\circ - 1080^\circ = -580^\circ. $$ For $$k = -2$$: $$ \theta = 500^\circ + 360^\circ(-2) = 500^\circ - 720^\circ = -220^\circ. $$ For $$k = -1$$: $$ \theta = 500^\circ + 360^\circ(-1) = 500^\circ - 360^\circ = 140^\circ. $$ **Final Answer** The angles coterminal to $$500^\circ$$ in the interval $$-720^\circ < \theta < 360^\circ$$ are: $$ -580^\circ,\quad -220^\circ,\quad 140^\circ. $$ The angles coterminal to $$500^\circ$$ within $$-720^\circ < \theta < 360^\circ$$ are $$-580^\circ$$, $$-220^\circ$$, and $$140^\circ$$.

\begin{tabular}{|l|} d) \( 490^{\circ} \) \\ \#3.) List all angles coterminal to \( 500^{\circ} \) where \( -720^{\circ}< \) \\ \( \theta<360^{\circ} \)\end{tabular}

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