Question

Use ubtraction Formula to write the expression as a trigonometric function of one number.(cos)(13pi)/15cos(-pi/5)-(sin)(13pi)/15sin(-pi/5)

Trigonometric Functions
ANSWERED
asked 2021-09-13
Use an Addition or Subtraction Formula to write the expression as a trigonometric function of one number, and then find its exact value.
\(\displaystyle{\left({\cos}\right)}\frac{{{13}\pi}}{{15}}{\cos{{\left(-\frac{\pi}{{5}}\right)}}}-{\left({\sin}\right)}\frac{{{13}\pi}}{{15}}{\sin{{\left(-\frac{\pi}{{5}}\right)}}}\)

Expert Answers (1)

2021-09-14
Our expression is
\(\displaystyle{\left({\cos}\right)}\frac{{{13}\pi}}{{15}}{\cos{{\left(-\frac{\pi}{{5}}\right)}}}-{\left({\sin}\right)}\frac{{{13}\pi}}{{15}}{\sin{{\left(-\frac{\pi}{{5}}\right)}}}\)
This kind of expression appears on the addition of angles formula for cosine
\(\displaystyle{\cos{{\left({x}+{y}\right)}}}={\cos{{x}}}{\cos{{y}}}-{\sin{{x}}}{\sin{{y}}}\)
with \(\displaystyle{x}=\frac{{{13}\pi}}{{15}},{y}=-\frac{\pi}{{5}}\)
So we have
\(\displaystyle{\left({\cos}\right)}\frac{{{13}\pi}}{{15}}{\cos{{\left(-\frac{\pi}{{5}}\right)}}}-{\left({\sin}\right)}\frac{{{13}\pi}}{{15}}{\sin{{\left(-\frac{\pi}{{5}}\right)}}}={\cos{{\left(\frac{{{13}\pi}}{{15}}+{\left(-\frac{\pi}{{5}}\right)}\right)}}}\)
\(\displaystyle={\cos{{\left(\frac{{{13}\pi}}{{15}}-\frac{{{3}\pi}}{{15}}\right)}}}\)
\(\displaystyle={\left({\cos}\right)}\frac{{{10}\pi}}{{15}}\)
\(\displaystyle={\left({\cos}\right)}\frac{{{2}\pi}}{{3}}\)
\(\displaystyle=-{\cos{{\left(\frac{\pi}{{3}}\right)}}}{\left[{\cos{{\left(\pi-{x}\right)}}}=-{\cos{{x}}}\right]}\)
\(\displaystyle=-\frac{{1}}{{2}}\)
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