The cosine function is a periodic function that describes the ratio of the adjacent side to the hypotenuse of a right triangle. It is one of the fundamental trigonometric functions, along with sine and tangent, and is widely used in various areas of mathematics, physics, and engineering.
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The cosine function is denoted by the symbol 'cos' and is one of the three primary trigonometric functions, along with sine and tangent.
The cosine function is closely related to the sine function, with the relationship cos(x) = sin(x + π/2).
The cosine function has a period of 2π, meaning it repeats its values every 2π units on the x-axis.
The graph of the cosine function is a sinusoidal curve that oscillates between -1 and 1 on the y-axis.
The cosine function is widely used in the analysis of periodic phenomena, such as oscillations, waves, and alternating current circuits.
Review Questions
Explain how the cosine function is related to the unit circle and the definition of a right triangle.
The cosine function is defined in terms of the unit circle, where the x-coordinate of a point on the circle corresponds to the cosine of the angle formed by that point and the positive x-axis. Specifically, the cosine of an angle θ is the ratio of the adjacent side to the hypotenuse of a right triangle with one angle equal to θ. This relationship between the cosine function and the geometry of a right triangle is fundamental to its applications in various areas of mathematics and physics.
Describe the key properties of the cosine function, such as its period, amplitude, and graph.
The cosine function has a period of 2π, meaning it repeats its values every 2π units on the x-axis. Its graph is a sinusoidal curve that oscillates between -1 and 1 on the y-axis, with a maximum value of 1 at 0, 2π, 4π, etc., and a minimum value of -1 at π, 3π, 5π, etc. The amplitude of the cosine function is 1, and it is a periodic, even function, meaning cos(-x) = cos(x).
Explain how the cosine function is used in the context of Taylor series expansions, specifically in the topic of 'Working with Taylor Series' (Section 6.4).
In the context of Taylor series expansions, the cosine function plays a crucial role. The Taylor series expansion of the cosine function is given by the formula: $$\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$$. This infinite series representation of the cosine function allows for the approximation of the function using a finite number of terms, which is particularly useful in the context of 'Working with Taylor Series' and the analysis of periodic phenomena.