Lesson Worksheet: Double-Angle and Half-Angle Identities Mathematics • 10th Grade

In this worksheet, we will practice using the double-angle and half-angle identities to evaluate trigonometric values.


Use the addition formula to find an expression for sin2๐›ผ.

  • Asincossin2๐›ผ=๐›ผ+๐›ผ๏Šจ๏Šจ
  • Bsincossin2๐›ผ=2๐›ผ๐›ผ
  • Csincossin2๐›ผ=๐›ผ+๐›ผ
  • Dsincossin2๐›ผ=๐›ผโˆ’๐›ผ๏Šจ๏Šจ
  • Esincossin2๐›ผ=๐›ผ๐›ผ


Find the value of 1โˆ’๏€ป๏‡1+๏€ป๏‡tantan๏Šจ๏Šญ๏Ž„๏Šฎ๏Šจ๏Šญ๏Ž„๏Šฎ without using a calculator.

  • Aโˆ’12
  • B1โˆš2
  • Cโˆ’1โˆš2
  • D12


Find the value of 27575sincosโˆ˜โˆ˜ without using a calculator.

  • Aโˆ’12
  • Bโˆš22
  • C12
  • Dโˆ’โˆš22


Which of the following is equal to โˆš1โˆ’2๐‘ฅcos?

  • Aโˆš2|๐‘ฅ|sin
  • B|๐‘ฅ|sin
  • C2|๐‘ฅ|sin
  • D2|๐‘ฅ|cos
  • Eโˆš2|๐‘ฅ|cos


Simplify 1โˆ’2๐‘ฅ1+2๐‘ฅcoscos.

  • Atan๏Šจ๐‘ฅ
  • Bcot2๐‘ฅ
  • Ctan๏Šจ2๐‘ฅ
  • Dcot๐‘ฅ


Find, without using a calculator, the value of sin2๐ด given cos๐ด=โˆ’1213 where 180โ‰ค๐ด<270โˆ˜โˆ˜.

  • A60169
  • Bโˆ’60169
  • C1013
  • D120169
  • Eโˆ’120169


Simplify sinsin๏Šจ๏Šช๐‘ฅโˆ’๐‘ฅ.

  • Asin2๐‘ฅ
  • Bcos2๐‘ฅ
  • C122๐‘ฅcos
  • D142๐‘ฅsin๏Šจ


Find the value of cos๏€ฝ๐œƒ2๏‰ given cos๐œƒ=1517 where 0<๐œƒ<90โˆ˜โˆ˜ without using a calculator.

  • A3โˆš55
  • B4โˆš1717
  • C3โˆš1010
  • Dโˆš1717


Using the half angle formulas, or otherwise, find the exact value of tan๏€ป๐œ‹8๏‡.

  • Aโˆ’1+โˆš2
  • B1+โˆš2
  • C12
  • Dโˆš2
  • Eโˆ’1โˆ’โˆš2


Find, without using a calculator, the value of sincos2๐ต22๐ต given cos๐ต=45 where 3๐œ‹2<๐ต<2๐œ‹.

  • Aโˆ’67
  • Bโˆ’625
  • Cโˆ’127
  • Dโˆ’1225

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