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Trigonometric Functions and Graphs

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Trigonometric functions such as sine, cosine, and tangent are fundamental in mathematics, revealing the relationship between angles and lengths. Their graphs display periodic patterns, characterized by features like amplitude and period. The sine and cosine functions oscillate with an amplitude of 1 and a period of 2π, while tangent has a period of π. Reciprocal functions and their inverses also play a crucial role in understanding trigonometry.

Exploring the Graphs of Trigonometric Functions

Trigonometric functions are a cornerstone of mathematical analysis, encapsulating the relationships between angles and lengths in right-angled triangles. The primary trigonometric functions—sine (sin), cosine (cos), and tangent (tan)—along with their reciprocals—cosecant (csc), secant (sec), and cotangent (cot)—exhibit periodic behavior that can be visualized through their graphs on a coordinate plane. These graphs are distinguished by features such as amplitude, the vertical stretch of the graph, and period, the horizontal length over which the function's values repeat. The amplitude is determined by the maximum vertical distance from the midline of the graph, while the period is the interval required for one complete cycle of the function's pattern.
Close-up view of a scientific calculator's matte gray trigonometric function keys, neatly arranged with subtle shadows, against a blurred background.

Characteristics of Trigonometric Function Graphs

The sine and cosine functions are characterized by an amplitude of 1, as they oscillate between -1 and 1. When expressed in the form y=a sin(bx) or y=a cos(bx), the amplitude is |a|, and the period is 2π/b, reflecting the frequency of oscillation. The tangent function, which does not have a maximum or minimum, has a period of π/b. The domain of sine and cosine encompasses all real numbers, with a range from -1 to 1, while tangent is defined for all real numbers except where it is undefined at odd multiples of π/2, and its range is all real numbers. The reciprocal functions, csc, sec, and cot, have domains that exclude the zeros of their corresponding functions and ranges that are all real numbers except the interval (-1, 1).

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00

In right-angled triangles, the primary ______ functions include sine, cosine, and tangent.

trigonometric

01

Sine/Cosine Range

Range of sine and cosine is from -1 to 1.

02

Tangent Period

Period of tangent function is π/b.

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