Question

COMPLETE THE SENTENCE Graphs of sine and cosine functions are called

Short answer

Waveforms or Waves

Step-by-step solution

Identify the Mathematical Term

Recall the term used to describe the graphs of sine and cosine functions. These types of graphs are typically smooth and continuous, moving in wave-like oscillations. Therefore, the correct term to complete the sentence is 'waveforms' or 'waves'.

Key concepts

Waveforms

When we talk about the graphs of sine and cosine functions, we often describe them as 'waveforms'. A waveform is a shape that represents how a signal moves or changes with time in a graphical format. In the case of sine and cosine functions, these waveforms show smooth and continuous curves that rise and fall in a pattern that repeats over a standard interval called the period. The peaks and troughs correspond to the maximum and minimum values of the functions, respectively.

Imagine the swell and dip of ocean waves; this visual is analogous to the sinusoidal waveforms produced by these trigonometric functions. The motion is rhythmic and predictable, which is why sine and cosine graphs are utilized to model periodic phenomena in real-world applications, such as sound waves, alternating current in electronics, and the motion of pendulums.

Trigonometric Functions

Trigonometric functions, specifically sine and cosine, are fundamental in mathematics for modeling periodic behavior. They are defined for any real number and are particularly useful for their periodic properties. The sine function, denoted as \( \text{sin} \), takes an angle as an input and outputs the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. The cosine function, \( \text{cos} \), similarly measures the ratio of the adjacent side to the hypotenuse.

The graphs of sine and cosine functions depict these ratios as they change with the angles, usually measured in radians. One full cycle of a sine or cosine function is completed in an interval of \( 2\text{π} \) radians, which corresponds to 360 degrees. Understanding the features of these functions such as amplitude (height of the waves), period (length of one full cycle), and phase (horizontal shift) is crucial for effectively using trigonometry to solve problems.

Oscillations

Oscillations refer to the repetitive variation, typically in time, of some measure as it swings back and forth between a minimum (trough) and a maximum (peak) value, with the sinusoidal path of the sine and cosine functions being prime examples. Within a trigonometric context, oscillation covers the rise and fall of the function values as they cycle through their periods.

An important characteristic of oscillation is the frequency, which is the number of cycles per unit time. For sine and cosine functions, the frequency is inversely proportional to the period; the longer the period, the lower the frequency, and vice versa. These concepts are not only relevant in math but are also found in a vast array of physical phenomena like the vibrations of strings, electricity and magnetism, and even circadian rhythms in biology.