Trigonometry
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Trigonometry Reference Home Back to Thinkquest Contents
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Definitions of Trigonometric Functions Trigonometric Functions of Sums and Differences Inverse Trigonometric Functions
| Definitions of Trigonometric Functions The above diagram shows the unit circle (a circle whose radius equals 1)including both the X and Y axes. Any point on a circle (P) that connects to the origin creates an angle X measured from the X-Axis. Sin(X) is defined as a line segment OA. It starts from the origin ending at the projection of the point P on the Y-Axis. Since OA cannot exceed the radius of the circle, Sin(X) varies between -1 and +1, corresponding to angles of 270 degrees, and 90 degrees. Cos(X) is defined as a line segment OB. It begins at the origin O, and ends at point B, which is the projection of point P on the X-Axis. Like the Sine function, the Cosine function is limited to -1 to +1. Tan(X) is defined as the line segment EC. Point E is the right most point on the circle on the X-Axis. A tangent line is drawn to the circle at that point. Point C is the intersection of this tangent line and the extended line OP. As angle X increases and approaches 90 degrees, the line segment OP becomes more and more parallel to the tangent line. As a result, the tangent increases to infinity because parallel lines never intersect. When X becomes 90 degrees, the lines are parallel and the tangent is undefined.
When the angle X is greater than 90 degrees, the point of intersection at C is below the X-Axis and the Tangent is therefore negative. The Tangent of X therefore varies from -infinity to +infinity. For X = 90 degrees or 270 degrees, the Tangent is undefined. Cot(X) is defined as the line segment FD. F is the upper most point on the circle on the Y-Axis. A tangent line is drawn on the circle at this point. Point D is the point of intersection of the extended line OP and this tangent line. Cot(X) also varies from -infinity to +infinity. When point P is on the X-Axis, and the angle is 0 or 180 degrees, the Cotangent is undefined. |
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