Question

Solve the equation: \cos\frac{23\pi}{12}+\cos\frac{5\pi}{12}

Trigonometry
ANSWERED
asked 2021-08-19
Solve the equation:
\(\displaystyle{\cos{{\frac{{{23}\pi}}{{{12}}}}}}+{\cos{{\frac{{{5}\pi}}{{{12}}}}}}\)

Expert Answers (1)

2021-08-20
Given
Using product sum dole
\(\displaystyle{\cos{{\left({A}\right)}}}+{\cos{{\left({B}\right)}}}={2}{\cos{{\left({\frac{{{A}+{B}}}{{{2}}}}\right)}}}{\cos{{\left({\frac{{{A}-{B}}}{{{2}}}}\right)}}}\)
\(\displaystyle{\cos{{\left({\frac{{{23}\pi}}{{{12}}}}\right)}}}+{\cos{{\left({\frac{{{5}\pi}}{{{12}}}}\right)}}}={2}{\cos{{\left({\frac{{{23}\pi+{5}\pi}}{{{24}}}}\right)}}}{\cos{{\left({\frac{{{\frac{{{23}\pi}}{{{12}}}}-{\frac{{{5}\pi}}{{{12}}}}}}{{{2}}}}\right)}}}\)
\(\displaystyle={2}{\cos{{\left({\frac{{{28}\pi}}{{{24}}}}\right)}}}{\cos{{\left({\frac{{{18}\pi}}{{{24}}}}\right)}}}\)
\(\displaystyle={2}{\cos{{\left({\frac{{{7}\pi}}{{{4}}}}\right)}}}{\cos{{\left({\frac{{{3}\pi}}{{{4}}}}\right)}}}\)
\(\displaystyle={2}{\left({\frac{{{1}}}{{\int{2}}}}\right)}{\left(-{\frac{{{1}}}{{\int{2}}}}\right)}\)
\(\displaystyle{\cos{{\left({\frac{{{23}\pi}}{{{12}}}}\right)}}}+{\cos{{\left({\frac{{{5}\pi}}{{{12}}}}\right)}}}=-{1}\)
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