| Summary: | Recent advances in mathematical inequalities suggest that bounds of polynomial-exponential-type are appropriate for evaluating key trigonometric functions. In this paper, we innovate in this sense by establishing new and sharp bounds of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><msup><mi>x</mi><mn>2</mn></msup><mo>)</mo></mrow><msup><mi>e</mi><mrow><mi>β</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></msup></mrow></semantics></math></inline-formula> for the trigonometric sinc and cosine functions. Our main result for the sinc function is a double inequality holding on the interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>π</mi><mo>)</mo></mrow></semantics></math></inline-formula>, while our main result for the cosine function is a double inequality holding on the interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>π</mi><mo>/</mo><mn>2</mn><mo>)</mo><mo>.</mo></mrow></semantics></math></inline-formula> Comparable sharp results for hyperbolic functions are also obtained. The proofs are based on series expansions, inequalities on the Bernoulli numbers, and the monotone form of the l’Hospital rule. Some comparable bounds of the literature are improved. Examples of application via integral techniques are given.
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