4 Elementary Functions Trigonometric Functions 4.15 Graphics 4.17 Special Values and Limits

§4.16 Elementary Properties

Notes:: See Hobson (1928, pp. 19–24).
Keywords:: trigonometric functions
Permalink:: http://dlmf.nist.gov/4.16

Figure 4.16.1: Quadrants for the angle $\theta$ .

Permalink:: http://dlmf.nist.gov/4.16.F1
Encodings:: pdf, png

Table 4.16.1: Signs of the trigonometric functions in the four quadrants.

Quadrant	$\mathop{\sin\/}\nolimits\theta,\mathop{\csc\/}\nolimits\theta$	$\mathop{\cos\/}\nolimits\theta,\mathop{\sec\/}\nolimits\theta$	$\mathop{\tan\/}\nolimits\theta,\mathop{\cot\/}\nolimits\theta$
I
II
III
IV

Symbols:: $\mathop{\csc\/}\nolimits z$ : cosecant function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cot\/}\nolimits z$ : cotangent function, $\mathop{\sec\/}\nolimits z$ : secant function, $\mathop{\sin\/}\nolimits z$ : sine function and $\mathop{\tan\/}\nolimits z$ : tangent function
A&S Ref:: 4.3.43
Referenced by:: §4.21(iii)
Permalink:: http://dlmf.nist.gov/4.16.T1

Table 4.16.2: Trigonometric functions: quarter periods and change of sign.

	$-\theta$	$\frac{1}{2}\pi\pm\theta$	$\pi\pm\theta$	$\frac{3}{2}\pi\pm\theta$	$2\pi\pm\theta$
$\mathop{\sin\/}\nolimits x$	$-\mathop{\sin\/}\nolimits\theta$	$\mathop{\cos\/}\nolimits\theta$	$\mp\mathop{\sin\/}\nolimits\theta$	$-\mathop{\cos\/}\nolimits\theta$	$\pm\mathop{\sin\/}\nolimits\theta$
$\mathop{\cos\/}\nolimits x$	$\mathop{\cos\/}\nolimits\theta$	$\mp\mathop{\sin\/}\nolimits\theta$	$-\mathop{\cos\/}\nolimits\theta$	$\pm\mathop{\sin\/}\nolimits\theta$	$\mathop{\cos\/}\nolimits\theta$
$\mathop{\tan\/}\nolimits x$	$-\mathop{\tan\/}\nolimits\theta$	$\mp\mathop{\cot\/}\nolimits\theta$	$\pm\mathop{\tan\/}\nolimits\theta$	$\mp\mathop{\cot\/}\nolimits\theta$	$\pm\mathop{\tan\/}\nolimits\theta$
$\mathop{\csc\/}\nolimits x$	$-\mathop{\csc\/}\nolimits\theta$	$\mathop{\sec\/}\nolimits\theta$	$\mp\mathop{\csc\/}\nolimits\theta$	$-\mathop{\sec\/}\nolimits\theta$	$\pm\mathop{\csc\/}\nolimits\theta$
$\mathop{\sec\/}\nolimits x$	$\mathop{\sec\/}\nolimits\theta$	$\mp\mathop{\csc\/}\nolimits\theta$	$-\mathop{\sec\/}\nolimits\theta$	$\pm\mathop{\csc\/}\nolimits\theta$	$\mathop{\sec\/}\nolimits\theta$
$\mathop{\cot\/}\nolimits x$	$-\mathop{\cot\/}\nolimits\theta$	$\mp\mathop{\tan\/}\nolimits\theta$	$\pm\mathop{\cot\/}\nolimits\theta$	$\mp\mathop{\tan\/}\nolimits\theta$	$\pm\mathop{\cot\/}\nolimits\theta$

Symbols:: $\mathop{\csc\/}\nolimits z$ : cosecant function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cot\/}\nolimits z$ : cotangent function, $\mathop{\sec\/}\nolimits z$ : secant function, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\tan\/}\nolimits z$ : tangent function and : real variable
A&S Ref:: 4.3.44
Permalink:: http://dlmf.nist.gov/4.16.T2

Table 4.16.3: Trigonometric functions: interrelations. All square roots have their principal values when the functions are real, nonnegative, and finite.

	$\mathop{\sin\/}\nolimits\theta=a$	$\mathop{\cos\/}\nolimits\theta=a$	$\mathop{\tan\/}\nolimits\theta=a$	$\mathop{\csc\/}\nolimits\theta=a$	$\mathop{\sec\/}\nolimits\theta=a$	$\mathop{\cot\/}\nolimits\theta=a$
$\mathop{\sin\/}\nolimits\theta$		$(1-a^{2})^{{1/2}}$	$a(1+a^{2})^{{-1/2}}$	$a^{{-1}}$	$a^{{-1}}(a^{2}-1)^{{1/2}}$	$(1+a^{2})^{{-1/2}}$
$\mathop{\cos\/}\nolimits\theta$	$(1-a^{2})^{{1/2}}$		$(1+a^{2})^{{-1/2}}$	$a^{{-1}}(a^{2}-1)^{{1/2}}$	$a^{{-1}}$	$a(1+a^{2})^{{-1/2}}$
$\mathop{\tan\/}\nolimits\theta$	$a(1-a^{2})^{{-1/2}}$	$a^{{-1}}(1-a^{2})^{{1/2}}$		$(a^{2}-1)^{{-1/2}}$	$(a^{2}-1)^{{1/2}}$	$a^{{-1}}$
$\mathop{\csc\/}\nolimits\theta$	$a^{{-1}}$	$(1-a^{2})^{{-1/2}}$	$a^{{-1}}(1+a^{2})^{{1/2}}$		$a(a^{2}-1)^{{-1/2}}$	$(1+a^{2})^{{1/2}}$
$\mathop{\sec\/}\nolimits\theta$	$(1-a^{2})^{{-1/2}}$	$a^{{-1}}$	$(1+a^{2})^{{1/2}}$	$a(a^{2}-1)^{{-1/2}}$		$a^{{-1}}(1+a^{2})^{{1/2}}$
$\mathop{\cot\/}\nolimits\theta$	$a^{{-1}}(1-a^{2})^{{1/2}}$	$a(1-a^{2})^{{-1/2}}$	$a^{{-1}}$	$(a^{2}-1)^{{1/2}}$	$(a^{2}-1)^{{-1/2}}$