Practice: Eulerโ€™s Formula for Trigonometric Identities

Use Eulerโ€™s formula to express tan(8๐œƒ) in terms of tan๐œƒ.

Hint: First write cos(8๐œƒ) and sin(8๐œƒ) in terms of sin๐œƒ and cos๐œƒ.

  • Atantantantantantantantantan(8๐œƒ)=8๐œƒโˆ’56๐œƒโˆ’56๐œƒ+8๐œƒ1+28๐œƒโˆ’70๐œƒ+28๐œƒ+๐œƒ๏Šฉ๏Šซ๏Šญ๏Šจ๏Šช๏Šฌ๏Šฎ
  • Btantantantantantantantantan(8๐œƒ)=8๐œƒโˆ’56๐œƒ+56๐œƒโˆ’8๐œƒ1โˆ’28๐œƒ+70๐œƒโˆ’28๐œƒ+๐œƒ๏Šฉ๏Šซ๏Šญ๏Šจ๏Šช๏Šฌ๏Šฎ
  • Ctantantantantantantantan(8๐œƒ)=โˆ’7๐œƒ+35๐œƒโˆ’21๐œƒ+๐œƒโˆ’1+21๐œƒโˆ’35๐œƒ+7๐œƒ๏Šฉ๏Šซ๏Šญ๏Šจ๏Šช๏Šฌ
  • Dtantantantantantantantantan(8๐œƒ)=โˆ’8๐œƒ+56๐œƒโˆ’56๐œƒ+8๐œƒ1โˆ’28๐œƒ+70๐œƒโˆ’28๐œƒ+๐œƒ๏Šฉ๏Šซ๏Šญ๏Šจ๏Šช๏Šฌ๏Šฎ
  • Etantantantantantantantan(8๐œƒ)=7๐œƒโˆ’35๐œƒ+21๐œƒโˆ’๐œƒ1โˆ’21๐œƒ+35๐œƒโˆ’7๐œƒ๏Šฉ๏Šซ๏Šญ๏Šจ๏Šช๏Šฌ

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