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The trigonometric functions sine, cosine, and tangent exhibit distinct symmetry properties that are reflected in their graphs. The sine and cosine functions are periodic, each with a period of \( 2\pi \) radians, which means their graphs repeat every \( 2\pi \) radians. The sine function is odd, meaning it has rotational symmetry about the origin, satisfying the identity \( \sin(-\theta) = -\sin(\theta) \). This property results in a graph that is symmetrical with respect to the origin. Conversely, the cosine function is even, displaying mirror symmetry about the y-axis, as illustrated by the identity \( \cos(-\theta) = \cos(\theta) \). This symmetry results in a cosine graph that is identical on either side of the y-axis.
The tangent function, with a period of \( \pi \) radians, is also odd like the sine function. Its graph shows rotational symmetry about the origin, adhering to the identity \( \tan(-\theta) = -\tan(\theta) \). However, unlike sine and cosine, the tangent function exhibits vertical asymptotes at odd multiples of \( \frac{\pi}{2} \), where the function approaches infinity. These symmetry properties and periodic behaviors are fundamental in understanding the trigonometric functions and their applications in various mathematical and real-world contexts.
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