1. **Stating the problem:** We have three right triangles with given sides and angles 30° and 60°, and we need to find missing sides using trigonometric ratios. 2. **Formulas and rules:** For a right triangle with angles 30° and 60°, the sides relate as follows: - $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ - $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$ - $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ For 30° and 60° angles: - $\tan(60^\circ) = \sqrt{3}$ - $\tan(30^\circ) = \frac{1}{\sqrt{3}}$ - $\cos(30^\circ) = \frac{\sqrt{3}}{2}$ 3. **Triangle a:** Given $a=11$, $b=5.2$, find $c$ and $\tan(60^\circ)$ check. Using $\tan(60^\circ) = \frac{b}{a}$: $$\tan(60^\circ) = \frac{5.2}{11} = 0.4727$$ But $\tan(60^\circ) = \sqrt{3} \approx 1.732$, so this is inconsistent. Instead, use $\tan(60^\circ) = \frac{b}{a}$ to find missing side $b$ if $a=11$: $$b = a \times \tan(60^\circ) = 11 \times \sqrt{3} = 11 \times 1.732 = 19.052$$ Given $b=5.2$ is likely a typo or different triangle. Find hypotenuse $c$ using Pythagoras: $$c = \sqrt{a^2 + b^2} = \sqrt{11^2 + 5.2^2} = \sqrt{121 + 27.04} = \sqrt{148.04} = 12.165$$ 4. **Triangle b:** Given $\text{adjacent} = 6.3$, hypotenuse $=9$, find opposite side $b$ and $\tan(60^\circ)$. Find opposite side $b$ using Pythagoras: $$b = \sqrt{9^2 - 6.3^2} = \sqrt{81 - 39.69} = \sqrt{41.31} = 6.426$$ Calculate $\tan(60^\circ) = \frac{b}{a} = \frac{6.426}{6.3} = 1.02$ (approx), close to $\sqrt{3}$ but slightly off. 5. **Triangle c:** Given $b=3.6$, $c=10.4$, and $a=16$, and $5\sqrt{3}$ (likely a side or value), find missing sides and verify. Check if $a=16$ is adjacent side, $b=3.6$ opposite, $c=10.4$ hypotenuse. Check Pythagoras: $$a^2 + b^2 = 16^2 + 3.6^2 = 256 + 12.96 = 268.96$$ $$c^2 = 10.4^2 = 108.16$$ Not equal, so $c$ is not hypotenuse here. If $c$ is hypotenuse, $c$ should be largest side, but $16 > 10.4$, so $16$ is hypotenuse. Check $c=5\sqrt{3} = 5 \times 1.732 = 8.66$ (approx), possibly a side length. Calculate $\cos(30^\circ) = \frac{4}{c}$ given: $$\cos(30^\circ) = \frac{\sqrt{3}}{2} = 0.866$$ So, $$c = \frac{4}{\cos(30^\circ)} = \frac{4}{0.866} = 4.618$$ 6. **Summary:** The problem data is inconsistent in places, but using trigonometric identities and Pythagoras theorem, missing sides can be found by: - Using $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ - Using Pythagoras: $c = \sqrt{a^2 + b^2}$ Final answers for triangle a: $$c = 12.165$$ For triangle b: $$b = 6.426$$ For triangle c, data inconsistent, but using $c = \frac{4}{\cos(30^\circ)} = 4.618$ if $4$ is adjacent side.