Examples of trigonometric functions applied to life are as follows:

Vibration damping. Many equipment and structures, such as bridges and high-rise buildings, need to resist the influence of earthquakes or other vibrations. By using trigonometric functions, engineers can design and adjust the vibration modes of these structures to reduce the influence of vibration. Radio communication. Radio signals will be affected by various factors in the transmission process, such as noise and interference.

By using trigonometric functions, engineers can design and optimize the coding methods of radio signals to improve the reliability and clarity of the signals. Music and sound processing. Trigonometric functions are also widely used in music and sound processing. For example, using sine function can simulate the sounds of various musical instruments, while using cosine function can create echo and reverberation effects.

At the same time, by analyzing the frequency spectrum of sound (trigonometric function will be used in frequency spectrum analysis), noise reduction and equalization of sound can be realized.

Expanding knowledge:

The early research on trigonometric functions can be traced back to ancient times. The founder of trigonometry in ancient Greece was Hipachus in the 2nd century BC. According to the practice of ancient Babylonians, he divided the circumference into 36 equal parts (that is, the arc of the circumference is 36 degrees, which is different from the modern arc system). For a given radian, he gives the corresponding chord length value, which is equivalent to the modern sine function.

hipparchus actually gave the earliest numerical table of trigonometric functions. However, trigonometry in ancient Greece was basically spherics. This is related to the fact that the main body of ancient Greek research is astronomy. Menelaus used sine to describe Menelaus theorem of spherical surface in his book Spherology. The application of trigonometry in ancient Greece and its astronomy reached its peak in Ptolemy era in Egypt.

Ptolemy calculated the sine values of 36-degree angle and 72-degree angle in Syntaxis Mathematica, and also gave the methods of calculating the sum angle formula and the half angle formula. Ptolemy also gave all sine values corresponding to all integer and semi-integer radians from to 18 degrees. Trigonometric function is one of the basic elementary functions, which takes angle (the most commonly used radian system in mathematics, the same below) as the independent variable.

The angle corresponds to the coordinate of the intersection point between the terminal edge of any angle and the unit circle or its ratio is a function of the dependent variable. It can also be equivalently defined by the lengths of various line segments related to the unit circle. Trigonometric function plays an important role in studying the properties of geometric shapes such as triangles and circles, and is also a basic mathematical tool for studying periodic phenomena. In mathematical analysis, trigonometric functions are also defined as infinite series or solutions of specific differential equations.

allow their values to be extended to any real value or even complex value. Common trigonometric functions include sine function, cosine function and tangent function. Other trigonometric functions, such as cotangent function, secant function, cotangent function, orthovector function, cofactor function, semiorthovector function and semicofactor function, are also used in other disciplines such as navigation, surveying and engineering.

the relationship between different trigonometric functions can be obtained by geometric intuition or calculation, which is called trigonometric identity. Trigonometric functions are generally used to calculate the sides and angles with unknown lengths in triangles, and are widely used in navigation, engineering and physics. In addition, using trigonometric functions as templates, we can define a class of similar functions, which are called hyperbolic functions.

Common hyperbolic functions are also called hyperbolic sine functions, hyperbolic cosine function and so on. Trigonometric function (also called circular function) is a function of angle; They are very important in studying triangles and modeling periodic phenomena and many other applications. Trigonometric function is usually defined as the ratio of two sides of a right triangle containing this angle.

can also be equivalently defined as the lengths of various line segments on the unit circle. More modern definitions express them as infinite series or solutions of specific differential equations, allowing them to be extended to any positive and negative values, even complex values.