Sine and cosine are two trigonometric functions commonly used in mathematics and physics. Sine (sin) is defined as the ratio of the length of the side opposite an angle in a right triangle to the length of the hypotenuse, while cosine (cos) is defined as the ratio of the length of the adjacent side to the hypotenuse.
The combination of sine and cosine functions allows for the creation of more complex functions that can better model real-world phenomena. By combining these functions, we can create functions with varying amplitudes, periods, and phases.
To combine sine and cosine functions, we use the sum and difference identities for sine and cosine. These identities allow us to express functions such as f(x) = sin(x) + cos(x) or g(x) = sin(x) - cos(x), which can be further manipulated and simplified.
f-g and f(x) - g(x) both represent the difference between two functions, but they are not the same. f-g represents the function obtained by subtracting g from f, while f(x) - g(x) represents the value of the difference between the two functions at a specific value of x.
Combining sine and cosine functions is useful in many fields, such as physics, engineering, and signal processing. These functions can be used to model periodic phenomena, such as sound and light waves, and to analyze and predict their behavior. They are also used in the design of electronic circuits and in the study of oscillatory motion.