1. Problem: Calculate the inverse tangent (arctan) values for given numbers and fractions. 2. Formula: The inverse tangent function is defined as $\tan^{-1}(x)$ which gives the angle whose tangent is $x$. 3. Important rule: The output angle is in radians or degrees depending on the calculator setting; here we use degrees. 4. Calculations: - a. $\tan^{-1}(0.725) \approx 36.01^\circ$ - b. $\tan^{-1}(0.325) \approx 18.00^\circ$ - c. $\tan^{-1}(\frac{3}{7}) = \tan^{-1}(0.4286) \approx 23.20^\circ$ - d. $\tan^{-1}(\frac{5}{12}) = \tan^{-1}(0.4167) \approx 22.62^\circ$ 5. Problem: Calculate angle $\angle E$ to the nearest degree given tangent values. 6. Formula: $\angle E = \tan^{-1}(\text{given tangent value})$ 7. Calculations: - a. $\angle E = \tan^{-1}(0.625) \approx 32^\circ$ - b. $\angle E = \tan^{-1}(0.812) \approx 39^\circ$ - c. $\angle E = \tan^{-1}(\frac{3}{5}) = \tan^{-1}(0.6) \approx 31^\circ$ - d. $\angle E = \tan^{-1}(\frac{7}{11}) = \tan^{-1}(0.6364) \approx 33^\circ$ 8. Problem: Calculate $\angle E$ in right triangles using tangent ratio. 9. Formula: $\tan(\angle E) = \frac{\text{opposite side}}{\text{adjacent side}}$, so $\angle E = \tan^{-1}(\frac{\text{opposite}}{\text{adjacent}})$. 10. Calculations: - a. Triangle JEI, right angle at J: Opposite = JE = 2.1 cm, Adjacent = JI = 3.7 cm $\angle E = \tan^{-1}(\frac{2.1}{3.7}) = \tan^{-1}(0.5676) \approx 29.54^\circ$ - b. Triangle GEF, right angle at F: Opposite = EF = 10 cm, Adjacent = EG = 18 cm $\angle E = \tan^{-1}(\frac{10}{18}) = \tan^{-1}(0.5556) \approx 29.05^\circ$ - c. Triangle CAE, right angle at C: Opposite = CE = 4 m, Adjacent = CA = 9 m $\angle E = \tan^{-1}(\frac{4}{9}) = \tan^{-1}(0.4444) \approx 23.96^\circ$ - d. Triangle LEM, right angle at M: Opposite = EM = 13.4 cm, Adjacent = LM = 16.1 cm $\angle E = \tan^{-1}(\frac{13.4}{16.1}) = \tan^{-1}(0.8323) \approx 39.87^\circ$ Final answers: - 19a: 36.01° - 19b: 18.00° - 19c: 23.20° - 19d: 22.62° - 20a: 32° - 20b: 39° - 20c: 31° - 20d: 33° - 21a: 29.54° - 21b: 29.05° - 21c: 23.96° - 21d: 39.87°