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The sine and cosine functions are commonly used to model periodic phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year.
The sine and cosine functions have several distinct characteristics: They are periodic functions with a period of 2π. The domain of each function is (− ∞, ∞) and the range is [− 1, 1]. The graph of y = sin x is symmetric about the origin, because it is an odd function.
Sine, Cosine and Tangent are the main functions used in Trigonometry and are based on a Right-Angled Triangle. Before getting stuck into the functions, it helps to give a name to each side of a right triangle: "Opposite" is opposite to the angle θ. "Adjacent" is adjacent to (next to) the angle θ. "Hypotenuse" is the long one.
In this section, we will interpret and create graphs of sine and cosine functions. Graphing Sine and Cosine Functions. Recall that the sine and cosine functions relate real number values to the x- and y-coordinates of a point on the unit circle. So what do they look like on a graph on a coordinate plane?
Determine amplitude, period, phase shift, and vertical shift of a sine or cosine graph from its equation. Graph variations of y=cos x and y=sin x . Determine a function formula that would have a given sinusoidal graph. Determine functions that model circular and periodic motion.
In fact Sine and Cosine are like good friends: they follow each other, exactly π /2 radians (90°) apart. Plot of the Tangent Function. The Tangent function has a completely different shape ... it goes between negative and positive Infinity, crossing through 0, and at every π radians (180°), as shown on this plot.
CHARACTERISTICS OF SINE AND COSINE FUNCTIONS. The sine and cosine functions have several distinct characteristics: They are periodic functions with a period of \(2\pi\). The domain of each function is \((−\infty,\infty)\) and the range is \([ −1,1 ]\). The graph of \(y=\sin\space x\) is symmetric about the origin, because it is an odd function.
The cosine and sine functions are called circular functions because their values are determined by the coordinates of points on the unit circle. For each real number \(t\), there is a corresponding arc starting at the point \((1, 0)\) of (directed) length \(t\) that lies on the unit circle.
The sine and cosine functions each vary in height, as their waves go up and down, between the y -values of −1 and +1. This value of " 1 " is called the "amplitude" of the waves. Note that the sine and cosine curves go one unit above and below their midlines; here, the midline happens to be the x -axis.
The Sine and Cosine Functions. Given an angle θ (in either degrees or radians) and the (x, y) coordinates of the corresponding point on the unit circle, we define cosine and sine as. cos(θ) = x and sin(θ) = y. Note that sine and cosine are functions that take angles as inputs.