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Sine is a fundamental trigonometric function that describes the relationship between the opposite side of a right-angled triangle and its hypotenuse. When dealing with an angle in standard position, where the vertex is at the origin and the initial side is along the positive x-axis, the sine of the angle is defined as the ratio of the y-coordinate of a point on the circle to the radius of the circle.

Here's how to grasp it with the example of the point \(7,7\):

• The y-coordinate is 7.
• The radius of the circle, using the Pythagorean theorem, is \(7\sqrt{2}\).

Thus, the sine of the angle is calculated as \( \sin \theta = \frac{y}{r} = \frac{7}{7\sqrt{2}} = \frac{\sqrt{2}}{2} \).

This concept helps us understand how a vertical position relates to the circle's radius. Therefore, sine values can range from -1 to 1, indicating the maximum and minimum positions on the unit circle.

Cosine is another critical trigonometric function. It measures the ratio of the adjacent side of a right triangle to its hypotenuse. For an angle in standard position, the cosine is the ratio of the x-coordinate of a point on the circle to the radius.

Let's break it down with the point \(7,7\):

• The x-coordinate is 7.
• The radius is again \(7\sqrt{2}\).

The cosine is then calculated as \( \cos \theta = \frac{x}{r} = \frac{7}{7\sqrt{2}} = \frac{\sqrt{2}}{2} \).

Cosine represents the horizontal aspect of the angle on the unit circle. Its values vary between -1 and 1, indicating how far left or right the point is from the origin. Understanding cosine lets us comprehend an angle's horizontal component effectively.

Tangent is a trigonometric function connecting sine and cosine, defined as the ratio of sine to cosine. It explains the slope of the line formed from the origin to a point on the circle. The tangent is calculated as the ratio of the y-coordinate to the x-coordinate.

Considering \(7,7\):

• Both coordinates are equal, y = x = 7.

Thus, \( \tan \theta = \frac{y}{x} = \frac{7}{7} = 1 \).

This value indicates a perfect 45-degree angle (or \( \frac{\pi}{4} \) radians) relative to the x-axis. Tangent can take any real number value and gives insight into the steepness or angle of inclination, making it useful in calculating real-world slopes and angles.