91Ó°ÊÓ

The cosine function is a fundamental trigonometric function that gives us the ratio of the adjacent side to the hypotenuse in a right triangle. Represented as \(\text{cos} \theta\), it is crucial for understanding angles and distances.
In the context of the unit circle, the cosine function provides the x-coordinate for a given angle. This means that for any angle \(\theta\), \(\text{cos} \theta\) tells you how far left or right you are from the center.
Remember, cosine values range between -1 and 1.
Key points of the cosine function include:
- \(\text{cos} 0 = 1\)
- \(\text{cos} \frac{\text{Ï€}}{2} = 0\)
- \(\text{cos} \text{Ï€} = -1\)

The unit circle is essential for understanding trigonometry. It is a circle with a radius of 1, centered at the origin of the coordinate system.
Angles in the unit circle are measured in radians. One complete revolution around the circle is \(\text{2Ï€}\) radians, which corresponds to 360 degrees.
Some key angles to remember in the unit circle include:
- 0 or \(2Ï€\) radians where \(\text{cos} = 1\)
- \(\frac{\text{Ï€}}{2}\) radians where \(\text{cos} = 0\)
- \(\text{Ï€}\) radians where \(cos = -1\)
The unit circle helps visualize why \(\text{cos}^{-1} 1\) results in 0 radians. This is because at 0 radians, the x-coordinate is 1.

Radians are a way of measuring angles based on the radius of a circle. One radian is the angle created when the length of the arc is equal to the radius of the circle.
There are \(\text{2Ï€}\) radians in a full circle, equivalent to 360 degrees. This means:
- \(\text{Ï€}\) radians is half a circle or 180 degrees.
- \(\frac{\text{Ï€}}{2}\) radians is a quarter circle or 90 degrees.
Using radians makes many mathematical formulas simpler and more elegant. For example, the length of an arc in a circle is just \(\text{rθ}\), where \(θ\) is in radians.
In our specific exercise, we find that \(\text{cos}^{-1} 1\) equals 0 radians because the angle whose cosine is 1 is exactly 0 radians on the unit circle.