TRIGONOMETRY TANθ WITH 360° VALUE WORKING MODEL

In trigonometry, the tangent function (tan) represents the ratio of the length of the side opposite a given angle to the length of the side adjacent to the angle in a right triangle. Like the sine and cosine functions, the tangent function is periodic with a period of 360 degrees (or 2π radians). Let’s explore how tan(θ) varies with different angles within a 360-degree cycle:

1. Understanding the Tangent Function:

• The tangent function is defined as tan(θ) = Opposite / Adjacent.
• In a unit circle (circle with radius 1), the tangent of an angle θ is equal to the y-coordinate divided by the x-coordinate of the point where the terminal side of the angle intersects the unit circle.

2. Graphing the Tangent Function:

• When graphed, the tangent function produces a repeating pattern of vertical asymptotes and sharp spikes as the angle θ increases from 0 to 360 degrees (or 0 to 2π radians).
• The function is undefined (approaches positive or negative infinity) at angles where the cosine (adjacent side) is zero, resulting in vertical asymptotes.
• Between the asymptotes, the tangent function oscillates between positive and negative values, depending on the quadrant.

3. Using Trigonometric Values:

• Trigonometric tables or calculators can provide the tangent values for specific angles within the 360-degree cycle.
• For example, tan(30°) = √3 / 3, tan(45°) = 1, tan(60°) = √3, tan(90°) is undefined (approaches positive infinity), and so on.

4. Applications:

• The tangent function is used in various fields, including geometry, physics, engineering, and astronomy.
• It describes phenomena such as slopes, angles of elevation and depression, and rotational motion.

In summary, the tangent function (tan) represents the ratio of the side opposite to the side adjacent to an angle in a right triangle, and it varies periodically with angles within a 360-degree cycle. Understanding the behavior of tan(θ) with different angles is fundamental in trigonometry and has practical applications in various fields.

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