Understanding the unit circle is essential for mastering trigonometry, particularly when it comes to angles and their corresponding sine, cosine, and tangent values. The unit circle is a circle with a radius of 1, centered at the origin of a coordinate system, and it provides a visual representation of these trigonometric functions.

To begin filling in the first quadrant of the unit circle, start with the key angle measures. The angles in degrees are 0°, 30°, 45°, 60°, and 90°, which correspond to the following radian measures: 0, \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), and \(\frac{\pi}{2}\). These relationships can be derived by recognizing that:

• 0° is at the point (1, 0).
• 90° is at the point (0, 1).
• 30° is \(\frac{1}{3}\) of 90°, leading to the radian measure of \(\frac{\pi}{6}\).
• 45° is halfway between 0° and 90°, resulting in \(\frac{\pi}{4}\).
• 60° is double the 30° angle, giving \(\frac{\pi}{3}\).

Next, the sine and cosine values for these angles can be determined using the square root method. For angles in the first quadrant, the sine (y-coordinate) and cosine (x-coordinate) values can be expressed as:

• For 0°: \((1, 0)\) → \(\cos(0) = 1\), \(\sin(0) = 0\)
• For 30°: \((\frac{\sqrt{3}}{2}, \frac{1}{2})\) → \(\cos(30) = \frac{\sqrt{3}}{2}\), \(\sin(30) = \frac{1}{2}\)
• For 45°: \((\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\) → \(\cos(45) = \frac{\sqrt{2}}{2}\), \(\sin(45) = \frac{\sqrt{2}}{2}\)
• For 60°: \((\frac{1}{2}, \frac{\sqrt{3}}{2})\) → \(\cos(60) = \frac{1}{2}\), \(\sin(60) = \frac{\sqrt{3}}{2}\)
• For 90°: \((0, 1)\) → \(\cos(90) = 0\), \(\sin(90) = 1\)

To find the tangent values, which are defined as the ratio of sine to cosine (or \(y/x\)), we can calculate:

• Tangent of 0°: \(\frac{0}{1} = 0\)
• Tangent of 30°: \(\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} \approx \frac{\sqrt{3}}{3}\) (after rationalizing the denominator)
• Tangent of 45°: \(\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1\)
• Tangent of 60°: \(\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}\)
• Tangent of 90°: \(\frac{1}{0}\) is undefined.

By practicing these calculations and familiarizing yourself with the unit circle, you can enhance your understanding of trigonometric functions and their applications. Repetition and reasoning through the relationships between angles, radians, and their corresponding sine, cosine, and tangent values will solidify your grasp of this fundamental concept in mathematics.