Angle Coterminal 1A7784

1. **State the problem:** Find angles with the same terminal side as given angles within specified ranges. 2. **Recall the formula:** Angles that differ by multiples of $360^\circ$ have the same terminal side. 3. **For (1):** Find all angles $\theta$ such that $\theta = 1050^\circ - 360^\circ k$ for integers $k$, and $-360^\circ \leq \theta \leq 720^\circ$. 4. Calculate: - For $k=2$: $1050^\circ - 360^\circ \times 2 = 1050^\circ - 720^\circ = 330^\circ$ (within range) - For $k=3$: $1050^\circ - 1080^\circ = -30^\circ$ (within range) - For $k=1$: $1050^\circ - 360^\circ = 690^\circ$ (within range) - For $k=0$: $1050^\circ$ (not in range, since $1050^\circ > 720^\circ$) - For $k=4$: $1050^\circ - 1440^\circ = -390^\circ$ (less than $-360^\circ$, exclude) So the angles are $-30^\circ$, $330^\circ$, and $690^\circ$. 5. **For (2):** Find the smallest positive angle coterminal with $1840^\circ$. Calculate $1840^\circ \mod 360^\circ$: $$1840^\circ - 360^\circ \times \left\lfloor \frac{1840}{360} \right\rfloor = 1840^\circ - 360^\circ \times 5 = 1840^\circ - 1800^\circ = 40^\circ$$ So the smallest positive angle is $40^\circ$. 6. **For (3):** Find the smallest positive angle coterminal with $-1840^\circ$. Calculate $-1840^\circ \mod 360^\circ$: Add $360^\circ$ multiples until positive: $$-1840^\circ + 360^\circ \times 6 = -1840^\circ + 2160^\circ = 320^\circ$$ So the smallest positive angle is $320^\circ$. **Final answers:** (1) $-30^\circ$, $330^\circ$, $690^\circ$ (2) $40^\circ$ (3) $320^\circ$