Use the Half-angle Formulas to find the exat value of each expression.

aflacatn 2021-08-22 Answered
Use the Half-angle Formulas to find the exat value of each expression.
\(\displaystyle{{\tan{{22.5}}}^{\circ}}\)

Want to know more about Trigonometry?

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question

Expert Answer

Margot Mill
Answered 2021-08-23 Author has 10015 answers
Trigonometry:
\(\displaystyle{\tan{{\left({22.5}^{\circ}\right)}}}={\tan{{t}}}\)
\(\displaystyle{\tan{{2}}}{t}={{\tan{{45}}}^{\circ}=}{1}\)
Use trig identify:
\(\displaystyle{\tan{{\left({2}{t}\right)}}}={\frac{{{2}{\tan{{t}}}}}{{{1}-{{\tan}^{{2}}{t}}}}}\)
\(\displaystyle{\frac{{{2}{\tan{{t}}}}}{{{1}-{{\tan}^{{2}}{t}}}}}={1}\)
\(\displaystyle{{\tan}^{{2}}{t}}+{2}{\tan{{t}}}-{1}={0}\)
\(\displaystyle{\tan{{t}}}={\frac{{-{2}\pm\sqrt{{{4}+{4}}}}}{{{2}}}}\)
\(\displaystyle={\frac{{-{2}\pm{2}\sqrt{{{2}}}}}{{{2}}}}\)
\(\displaystyle=-{1}\pm\sqrt{{{2}}}\)
Since \(\displaystyle{\tan{{22.5}}}\) is positive, then take the positive answer:
\(\displaystyle{\tan{{\left({22.5}\right)}}}=-{1}+\sqrt{{{2}}}\)
Have a similar question?
Ask An Expert
18
 

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more
...