How do you find the exact value of arctan (2)?

The **exact value** of arctan(2) cannot be expressed as a simple fraction or a well-known angle in degrees or radians (like π/4, π/6, etc.), because 2 is not one of the special values for which we have a neat trigonometric or inverse trigonometric result. However, we can provide a detailed explanation of what **arctan(2)** represents and how you might evaluate it approximately:

---

### Understanding arctan(2)

1. **Definition**: The arctangent function, written as arctan(x), is the inverse of the tangent function. For a given value `y = arctan(x)`, this means:
- `tan(y) = x`, where y lies within the interval (-π/2, π/2).

In this case, we are solving for y such that:
```
tan(y) = 2
```
and y is in the range (-π/2, π/2).

2. **Exact Value**: The value of arctan(2) does not correspond to a simple multiple of π or a fraction of π. Hence, there is no "exact" symbolic answer beyond simply writing it as arctan(2).

---

### Approximating arctan(2)

1. **Numerical Approximation**:
To approximate arctan(2), you can use a calculator or numerical techniques. Using a calculator:
```
arctan(2) ≈ 1.107 radians
```
In degrees:
```
arctan(2) ≈ 63.43 degrees
```

2. **Series Expansion**:
The arctan function can be expressed using an infinite series:
```
arctan(x) = x - (x^3)/3 + (x^5)/5 - (x^7)/7 + ...
```
Substituting x = 2 into this series:
```
arctan(2) ≈ 2 - (2^3)/3 + (2^5)/5 - (2^7)/7 + ...
= 2 - 8/3 + 32/5 - 128/7 + ...
```
Computing the first few terms:
```
arctan(2) ≈ 2 - 2.67 + 6.4 - 18.29 + ...
≈ 1.107 (approximated value)
```

---

### Geometric Interpretation

- The value of arctan(2) corresponds to the angle whose tangent is 2. If you visualize this on a right triangle:
- The side opposite the angle is 2 units long.
- The side adjacent to the angle is 1 unit long.
- The hypotenuse can be found using the Pythagorean theorem: sqrt(2^2 + 1^2) = sqrt(5).
- The angle is arctan(2).

---

### Conclusion

While the exact symbolic value of arctan(2) cannot be simplified further, it can be approximated numerically as:
- Approximately **1.107 radians**, or **63.43 degrees**.
For further precision, use a calculator or software.