Digital Library of Mathematical Functions
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4 Elementary FunctionsTrigonometric Functions

§4.18 Inequalities

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Notes:
For (4.18.1) see Copson (1935, p. 136). (4.18.3) follows by the same method and (4.18.2) is a consequence. (4.18.5) to (4.18.9) are straightforward and (4.18.10) is obtained from the Maclaurin expansions of \mathop{\cos\/}\nolimits z and \mathop{\sin\/}\nolimits z. For the second inequality in (4.18.4), it is sufficient to show f(x)\equiv 4x(1-x)-\mathop{\sin\/}\nolimits\!\left(\pi x\right)\geq 0 for 0\leq x\leq 1/2. The function f(x) is zero at x=0 and x=1/2 and it has no zeros in (0,1/2), because f^{{\prime}}(x)=4(1-2x)-\pi\mathop{\cos\/}\nolimits\!\left(\pi x\right) can have only one zero in (0,1/2) where y=4(1-2x) intersects y=\pi\mathop{\cos\/}\nolimits\!\left(\pi x\right). The first inequality is proved similarly.
Keywords:
inequalities, trigonometric functions
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http://dlmf.nist.gov/4.18

Jordan’s Inequality

4.18.1\frac{2x}{\pi}\leq\mathop{\sin\/}\nolimits x\leq x,0\leq x\leq\frac{1}{2}\pi.
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Symbols:
\mathop{\sin\/}\nolimits z: sine function and x: real variable
A&S Ref:
4.3.79 4.3.80
Referenced by:
§4.18
Permalink:
http://dlmf.nist.gov/4.18.E1
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4.18.2x\leq\mathop{\tan\/}\nolimits x,0\leq x<\frac{1}{2}\pi,
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Symbols:
\mathop{\tan\/}\nolimits z: tangent function and x: real variable
A&S Ref:
4.3.80
Referenced by:
§4.18
Permalink:
http://dlmf.nist.gov/4.18.E2
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4.18.3\mathop{\cos\/}\nolimits x\leq\frac{\mathop{\sin\/}\nolimits x}{x}\leq 1,0\leq x\leq\pi,
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Symbols:
\mathop{\cos\/}\nolimits z: cosine function, \mathop{\sin\/}\nolimits z: sine function and x: real variable
A&S Ref:
4.3.81
Referenced by:
§4.18
Permalink:
http://dlmf.nist.gov/4.18.E3
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4.18.4\pi<\frac{\mathop{\sin\/}\nolimits\!\left(\pi x\right)}{x(1-x)}\leq 4,0<x<1.
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Symbols:
\mathop{\sin\/}\nolimits z: sine function and x: real variable
A&S Ref:
4.3.82
Referenced by:
§4.18
Permalink:
http://dlmf.nist.gov/4.18.E4
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With z=x+iy,

4.18.8|\mathop{\cos\/}\nolimits z|\leq\mathop{\cosh\/}\nolimits|z|,
4.18.9|\mathop{\sin\/}\nolimits z|\leq\mathop{\sinh\/}\nolimits|z|,
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Symbols:
\mathop{\sinh\/}\nolimits z: hyperbolic sine function, \mathop{\sin\/}\nolimits z: sine function and z: complex variable
A&S Ref:
4.3.87
Referenced by:
§4.18
Permalink:
http://dlmf.nist.gov/4.18.E9
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4.18.10
|\mathop{\cos\/}\nolimits z|<2,
|\mathop{\sin\/}\nolimits z|\leq\tfrac{6}{5}|z|,|z|<1.
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Symbols:
\mathop{\cos\/}\nolimits z: cosine function, \mathop{\sin\/}\nolimits z: sine function and z: complex variable
A&S Ref:
4.3.88
Referenced by:
§4.18
Permalink:
http://dlmf.nist.gov/4.18.E10
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For more inequalities see Mitrinović (1964, pp. 101–111), Mitrinović (1970, pp. 235–265), and Bullen (1998, pp. 250–254).

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