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Trigonometry delves into the relationships between triangle angles and sides, focusing on sine, cosine, and tangent functions. These functions exhibit periodic behavior and are graphically represented to aid in solving equations. The sine and cosine functions oscillate between -1 and 1, while the tangent function's range is all real numbers. Understanding their graphical properties is crucial for finding solutions within specific intervals.
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The sine function relates the size of an angle to the ratios of a triangle's sides and is characterized by its periodic, sinusoidal graph
The cosine function is similar to the sine function but is phase-shifted and exhibits symmetry properties, making it useful for solving equations
The tangent function has a unique graph with a period of π radians and an unbounded range, making it useful for solving equations with multiple solutions
The sine and cosine functions exhibit symmetry properties, allowing for the identification of additional solutions to equations
Trigonometric functions are inherently periodic, meaning their values repeat in regular intervals, which is crucial for solving equations
The tangent function has vertical asymptotes at odd multiples of π/2 radians, where it approaches infinity, making it useful for solving equations with unbounded solutions
Graphical methods involve plotting the trigonometric function and the equation's constant value on the same coordinate plane to identify solutions
Symmetry properties of trigonometric functions can be leveraged to identify additional solutions to equations
Understanding the periodic nature of trigonometric functions is essential for accurately solving equations within specific ranges
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Trigonometric functions periodicity
Sine, cosine, and tangent functions repeat values at regular intervals; sine and cosine have a period of 2π, tangent has a period of π.
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Sine and cosine value range
Sine and cosine functions oscillate between -1 and 1, representing their maximum and minimum values respectively.
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Tangent function value range
Tangent function relates opposite to adjacent side lengths in a right-angled triangle and can take any real number as its value, unlike sine and cosine.
