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Determine the exact value of the real number t. just give a very short solution and exact answer

The point on the unit circle is given as \( \left( \frac{4}{5}, -\frac{3}{5} \right) \). 

The \( x \)-coordinate \( \frac{4}{5} \) represents \( \cos(\theta) \) and the \( y \)-coordinate \( -\frac{3}{5} \) represents \( \sin(\theta) \).

Now, to find \( t \), which is the angle \( \theta \) in radians, we use the arctangent function:
\[
\theta = \tan^{-1} \left( \frac{\sin(\theta)}{\cos(\theta)} \right) = \tan^{-1} \left( \frac{-\frac{3}{5}}{\frac{4}{5}} \right) = \tan^{-1} \left( -\frac{3}{4} \right)
\]
Thus, the angle \( \theta \) is approximately \( -\tan^{-1}(0.75) \), or exactly:
\[
t = -\frac{\pi}{4}
\] 

Would you like more details or have any questions?

Here are five related questions:
1. How can you convert this angle to degrees?
2. What is the general formula for tangent in terms of sine and cosine?
3. How can you find the reference angle if given an arbitrary point on the unit circle?
4. What are the sine and cosine values of \( \frac{\pi}{4} \)?
5. How does the unit circle help in solving trigonometric equations?

**Tip**: Always remember that angles in the third and fourth quadrants give negative sine values.

Determine the exact value of the real number t.

Find the exact value of the real number t given a point on the unit circle. This problem involves using trigonometric functions such as sine, cosine, and tangent, with a step-by-step solution involving inverse trigonometric functions.

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