+ We a colcelaton ion and dhe volve of e in the givation belowe 1. $$ \sin heta=0.78 $$ 2. $$ an heta=3 \sin heta 1 $$ 3. $$ 3 \sin heta=2 $$ 4. $$ \sin heta= an 21+\cos 72 $$

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+ We a colcelaton ion and dhe volve of e in the givation belowe 1. $$ \sin 	heta=0.78 $$ 2. $$ 	an 	heta=3 \sin 	heta 1 $$ 3. $$ 3 \sin 	heta=2 $$ 4. $$ \sin 	heta=	an 21+\cos 72 $$

UpStudy AI · Accepted Answer

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:
  $$\cos\left(72\right)$$
Calculate or simplify the expression $$ \tan(21) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\tan\left(21\right)$$
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
Now, let's summarize the results from all equations:

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 values.

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:
$$
\theta = \arcsin(0.78)
$$

This value can be expressed in degrees. However, I will provide the numerical value for $$ \theta $$ in degrees.

Let's calculate $$ \theta $$ in degrees.
Calculate the value by following steps:
- step0: Calculate:
  $$\arcsin\left(0.78\right)$$
The value of $$ \theta $$ is approximately:
$$
\theta \approx 0.894666 \text{ radians}
$$

To convert this to degrees:
$$
\theta \approx 0.894666 \times \frac{180}{\pi} \approx 51.24^\circ
$$

### Final Answer
The value of $$ \theta $$ is approximately:
$$
\theta \approx 51.24^\circ
$$
$$ \theta \approx 51.24^\circ $$

+ 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. \( \sin \theta=\tan 21+\cos 72 \)

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