Worksheet: Double-Angle and Half-Angle Identities

In this worksheet, we will practice using the Pythagorean identity and double-angle formulas to evaluate trigonometric values.

Q1:

Find the value of cos2๐ด given cos๐ด=โˆ’35 where 90<๐ด<180โˆ˜โˆ˜ without using a calculator.

  • Aโˆ’725
  • B2425
  • Cโˆ’2425
  • D725

Q2:

Find the value of cos๏€ผ๐ด2๏ˆ given sin๏€ผ๐ด2๏ˆ=35 without using a calculator.

  • Aโˆ’34
  • B45
  • C34
  • Dโˆ’45

Q3:

Find, without using a calculator, the value of sin2๐ด given tan๐ด=โˆ’512 where 3๐œ‹2<๐ด<2๐œ‹.

  • Aโˆ’119169
  • Bโˆ’60169
  • Cโˆ’120169
  • D119169

Q4:

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

Q5:

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

Q6:

Find, without using a calculator, the value of sin๏€ฝ๐œƒ2๏‰ given tan๐œƒ=โˆ’158 where 3๐œ‹2<๐œƒ<2๐œ‹.

  • A5โˆš3434
  • Bโˆš2626
  • C3โˆš3434
  • D4โˆš1717

Q7:

๐ด๐ต๐ถ is a triangle where tan๐ถ=815. Find the value of sin๏€ผ๐ด+๐ต2๏ˆ.

  • A5โˆš34
  • B8โˆš17
  • C4โˆš17
  • D8โˆš15

Q8:

Find the value of cos(๐œ‹+2๐ด) given sin(270+๐ด)=โˆ’1517โˆ˜ where 3๐œ‹2<๐ด<2๐œ‹.

  • A161289
  • Bโˆ’161289
  • Cโˆ’240289
  • D240289

Q9:

Find, without using a calculator, 1โˆ’2๐‘‹1+2๐‘‹coscos given tan๐‘‹=4 where ๐‘‹โˆˆ๏€ผ๐œ‹,3๐œ‹2๏ˆ.

  • Aโˆ’116
  • B116
  • Cโˆ’16
  • D16

Q10:

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

  • Asincos๐‘ฅโˆ’๐‘ฅ
  • Bcossin๐‘ฅโˆ’๐‘ฅ
  • C|๐‘ฅโˆ’๐‘ฅ|cossin
  • D|๐‘ฅ+๐‘ฅ|cossin
  • Ecossin๐‘ฅ+๐‘ฅ

Q11:

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

Q12:

Find the value of 1+2๐ด1+2๐ดsincos given tan๐ด=526 where 0<๐ด<๐œ‹3 without using a calculator.

  • Aโˆ’9611,352
  • B1,352961
  • C9611,352
  • Dโˆ’1,352961

Q13:

Find the value of tancot15730โ€ฒ+15730โ€ฒโˆ˜โˆ˜ and then tancot๏Šจโˆ˜๏Šจโˆ˜15730โ€ฒ+15730โ€ฒ without using a calculator.

  • Aโˆ’2โˆš2, 6
  • Bโˆš2, 6
  • Cโˆ’2โˆš2, 8
  • D2โˆš2, 16

Q14:

Find the value of sin4๐‘‹ given 2๐‘‹๐‘‹โˆ’2๐‘‹๐‘‹=926sincoscossin๏Šฉ๏Šฉ.

  • A913
  • Bโˆ’913
  • Cโˆ’926
  • D926

Q15:

Find, without using a calculator, the value of tan4๐‘‹ given sincos๐‘‹๐‘‹=โˆ’14 where ๐‘‹โˆˆ๏€ผ๐œ‹2,3๐œ‹4๏ˆ.

  • A1โˆš3
  • Bโˆ’1โˆš3
  • Cโˆ’โˆš3
  • Dโˆš3

Q16:

Use the addition formula to find an expression for cos2๐›ผ.

  • Acoscossin2๐›ผ=2๐›ผ+2๐›ผ
  • Bcoscossin2๐›ผ=๐›ผโˆ’๐›ผ๏Šจ๏Šจ
  • Ccoscossin2๐›ผ=๐›ผ+๐›ผ๏Šจ๏Šจ
  • Dcoscossin2๐›ผ=2๐›ผโˆ’2๐›ผ
  • Ecossincos2๐›ผ=2๐›ผ๐›ผ

Q17:

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

Q18:

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

Q19:

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

Q20:

Evaluate 3611230โ€ฒ1โˆ’11230โ€ฒtantanโˆ˜๏Šจโˆ˜ without using a calculator.

  • A18
  • B36
  • Cโˆ’18
  • D1

Q21:

Simplify tantan3544โ€ฒ14โ€ฒโ€ฒ1โˆ’3544โ€ฒ14โ€ฒโ€ฒโˆ˜๏Šจโˆ˜.

  • Atan3544โ€ฒ14โ€ฒโ€ฒ2โˆ˜
  • Btan7128โ€ฒ28โ€ฒโ€ฒโˆ˜
  • Ctan7128โ€ฒ28โ€ฒโ€ฒ2โˆ˜
  • Dtan๏Šจโˆ˜7128โ€ฒ28โ€ฒโ€ฒ2

Q22:

๐ด๐ต๐ถ is a triangle, where the ratio between its lengths ๐‘Ž, ๐‘, and ๐‘ is 4โˆถ3โˆถ5. Find tan2๐ด.

  • A2425
  • B127
  • C247
  • Dโˆ’247
  • Eโˆ’127

Q23:

Find the value of tancot15730โ€ฒ+15730โ€ฒโˆ˜โˆ˜ without using a calculator.

  • Aโˆš2
  • Bโˆ’2โˆš2
  • C2โˆš2
  • Dโˆ’โˆš2

Q24:

Calculate cossinsincos๏Šฌโˆ˜๏Šฌโˆ˜๏Šจโˆ˜๏Šจโˆ˜15โˆ’151โˆ’1515.

  • Aโˆ’12
  • B12
  • Cโˆš32
  • Dโˆ’โˆš32

Q25:

Find the value of tan๐œƒ given sectan๐œƒโˆ’๐œƒ=34 where 0<๐œƒ<๐œ‹2.

  • A258
  • B2524
  • C318
  • D724

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.