+ We a colcelaton ion and dhe volve of e in the givation belowe 1. \( \sin \theta=0.78 \) 2. \( \tan \theta=3 \sin \theta 1 \) 3. \( 3 \sin \theta=2 \) 4. \( \sin \theta=\tan 21+\cos 72 \)

Let's analyze the given equations step by step to find the value of \( \theta \). ### Known Conditions 1. \( \sin \theta = 0.78 \) 2. \( \tan \theta = 3 \sin \theta + 1 \) 3. \( 3 \sin \theta = 2 \) 4. \( \sin \theta = \tan 21 + \cos 72 \) ### Step 1: Solve Each Equation #### Equation 1: \( \sin \theta = 0.78 \) This gives us a direct value for \( \sin \theta \). #### Equation 2: \( \tan \theta = 3 \sin \theta + 1 \) Substituting \( \sin \theta = 0.78 \): \[ \tan \theta = 3(0.78) + 1 \] #### Equation 3: \( 3 \sin \theta = 2 \) This can be solved for \( \sin \theta \): \[ \sin \theta = \frac{2}{3} \approx 0.6667 \] #### Equation 4: \( \sin \theta = \tan 21 + \cos 72 \) We need to calculate \( \tan 21 \) and \( \cos 72 \). ### Step 2: Calculate Values for Equation 4 Let's calculate \( \tan 21 \) and \( \cos 72 \). 1. **Calculate \( \tan 21 \)** 2. **Calculate \( \cos 72 \)** I'll perform these calculations now. Calculate the value by following steps: - step0: Calculate: \(\tan\left(21\right)\) Calculate or simplify the expression \( \cos(72) \). Calculate the value by following steps: - step0: Calculate: \(\cos\left(72\right)\) The calculations yield the following results: 1. \( \tan(21) \approx 0.383864 \) 2. \( \cos(72) \approx 0.309017 \) ### Step 3: Substitute into Equation 4 Now we can substitute these values into Equation 4: \[ \sin \theta = \tan(21) + \cos(72) \approx 0.383864 + 0.309017 \] Let's calculate this sum. Calculate the value by following steps: - step0: Calculate: \(0.383864+0.309017\) - step1: Add the numbers: \(0.692881\) The sum gives us: \[ \sin \theta \approx 0.692881 \] ### Summary of Results 1. From Equation 1: \( \sin \theta = 0.78 \) 2. From Equation 2: \( \tan \theta = 3(0.78) + 1 \) 3. From Equation 3: \( \sin \theta = \frac{2}{3} \approx 0.6667 \) 4. From Equation 4: \( \sin \theta \approx 0.692881 \) ### Conclusion The values of \( \sin \theta \) from the different equations are not consistent. The first equation gives \( \sin \theta = 0.78 \), while the others yield different results. To find a consistent value for \( \theta \), we can use the first equation \( \sin \theta = 0.78 \) and calculate \( \theta \) using the inverse sine function: \[ \theta = \arcsin(0.78) \] Let's calculate \( \theta \). Calculate the value by following steps: - step0: Calculate: \(\arcsin\left(0.78\right)\) The calculation for \( \theta \) gives us: \[ \theta = \arcsin(0.78) \] To find the numerical value, we can evaluate \( \arcsin(0.78) \) in degrees. The approximate value is: \[ \theta \approx 51.06^\circ \] ### Final Answer The value of \( \theta \) is approximately \( 51.06^\circ \).