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Determine the exact value of the real number t

The image shows a unit circle with a point marked on it, and the coordinates of this point are given as \( \left( \frac{12}{13}, \frac{5}{13} \right) \). The angle \( \theta \) is also indicated, and we are asked to find the value of the real number \( t \), which seems to represent \( \theta \) or a related quantity.

In the context of the unit circle, we know that the coordinates \( (x, y) \) of a point on the circle can be written as \( (\cos(\theta), \sin(\theta)) \), where \( \theta \) is the angle measured counterclockwise from the positive \( x \)-axis.

Given the coordinates \( \left( \frac{12}{13}, \frac{5}{13} \right) \), we can infer the following:
- \( \cos(\theta) = \frac{12}{13} \)
- \( \sin(\theta) = \frac{5}{13} \)

Now, to find \( t \), which could represent the angle \( \theta \), we can use the inverse trigonometric functions. Specifically, we can find \( \theta \) using the arctangent function because:

\[
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{5}{13}}{\frac{12}{13}} = \frac{5}{12}
\]

Thus, \( \theta = \arctan\left( \frac{5}{12} \right) \).

Finally, the exact value of \( t \) is \( \arctan\left( \frac{5}{12} \right) \), which can be expressed in radians if needed, but this is the exact form of the angle.

Would you like further details, or do you have any questions?

Here are some additional questions to explore:
1. How would you convert this angle \( \theta \) from radians to degrees?
2. What are the sine and cosine of the angle in terms of special triangles or reference angles?
3. How would you verify that the point \( \left( \frac{12}{13}, \frac{5}{13} \right) \) lies on the unit circle?
4. What is the significance of the unit circle in trigonometry?
5. How does the arctangent function relate to the tangent function geometrically?

**Tip:** The tangent of an angle can be found using the ratio of the sine to the cosine of that angle.

Determine the exact value of the real number t, based on the given unit circle diagram with a point having coordinates (12/13, 5/13).

Explore how to determine the exact value of the angle θ based on the unit circle coordinates (12/13, 5/13). This problem covers core trigonometric concepts including sine, cosine, and the arctangent function, suitable for high school students.

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