<p>We present several inequalities for sums involving the modulus of the sine function. One of our results states that for all integers <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n\ge 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and real numbers <i>x</i>, we have <Equation ID="Equ42"> <EquationSource Format="TEX">\( \sum _{k=1}^n (n-k+1)(n-k+2)|\sin (kx)| \le C (n+1)^3 \)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <munderover> <mo>∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>-</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>-</mo> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">|</mo> <mo>sin</mo> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>C</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>3</mn> </msup> </mrow> </math></EquationSource> </Equation>with the best possible constant factor <Equation ID="Equ43"> <EquationSource Format="TEX">\( C=\frac{1}{432} \bigl (9+\sqrt{41}\bigl ) \sqrt{14+6\sqrt{41}}=0.25814... . \)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>C</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>432</mn> </mfrac> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mn>9</mn> <mo>+</mo> <msqrt> <mn>41</mn> </msqrt> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <msqrt> <mrow> <mn>14</mn> <mo>+</mo> <mn>6</mn> <msqrt> <mn>41</mn> </msqrt> </mrow> </msqrt> <mo>=</mo> <mn>0.25814</mn> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </math></EquationSource> </Equation></p>