Cosine Function

Definition

Definition of the Cosine Function

In a right triangle, let $\theta$ be an acute angle, then cosine is defined as:

\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}

where “adjacent” refers to the side adjacent to angle $\theta$ , and “hypotenuse” is the longest side of the triangle.

符号说明

Symbol	Type	Pronunciation/Explanation	Meaning in This Definition
$\theta$	Greek letter	Theta (theta)	Represents the measure of an angle (acute angle)

Find the domain and range of $y = \cos x$ .

Answer and Explanation (3 个标签)

cosine function domain range

Domain: $\mathbb{R}$ ; Range: $[-1, 1]$ .

Determine the periodicity and parity of $y = \cos x$ .

Answer and Explanation (3 个标签)

cosine function period parity

Period is $2\pi$ , it is an even function.

Sketch the approximate graphs of $y = \sin x$ and $y = \cos x$ , and compare their periodicity and symmetry.

Answer and Explanation (4 个标签)

sine function cosine function period parity

Both $y = \sin x$ and $y = \cos x$ have period $2\pi$ . $\sin x$ is an odd function, $\cos x$ is an even function.

Which of the following functions are odd functions?
(A) $y = \sin x$
(B) $y = \cos x$
(C) $y = \tan x$
(D) $y = x^2$

Answer and Explanation (5 个标签)

sine function cosine function tangent function odd function even function

(A), (C) are odd functions, (B), (D) are not.

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$\pi$	Greek letter	Pi (pie)	Pi, used to represent the period of the cosine function ( $2\pi$ )

Chinese Term	English Term	Phonetic	Explanation
余弦函数	cosine function	/ˈkəʊsaɪn ˈfʌŋkʃən/	One of the trigonometric functions, denoted as $\cos x$
余弦	cosine	/ˈkəʊsaɪn/	Function name, abbreviated as $\cos$
邻边	adjacent side	/əˈdʒeɪsənt saɪd/	The right side adjacent to the angle in a right triangle
斜边	hypotenuse	/haɪˈpɒtənjuːz/	The longest side in a right triangle
周期	period	/ˈpɪəriəd/	The smallest interval at which function values repeat
定义域	domain	/dəʊˈmeɪn/	The range of values for the independent variable
值域	range	/reɪndʒ/	The range of values for the function
偶函数	even function	/ˈiːvən ˈfʌŋkʃən/	A function satisfying $f(-x) = f(x)$

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