Lesson Worksheet: Double-Angle and Half-Angle Identities Mathematics

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

Q1:

Find the value of cos2𝐴 given cos𝐴=35 where 90<𝐴<180 without using a calculator.

  • A725
  • B2425
  • C2425
  • D725

Q2:

Find the value of cos𝐴2 given sin𝐴2=35 without using a calculator.

  • A34
  • B45
  • C34
  • D45

Q3:

Find, without using a calculator, the value of sin2𝐴 given tan𝐴=512 where 3𝜋2<𝐴<2𝜋.

  • A119169
  • B60169
  • C120169
  • D119169

Q4:

Find the value of cos𝜃2 given cos𝜃=1517 where 0<𝜃<90 without using a calculator.

  • A355
  • B41717
  • C31010
  • D1717

Q5:

Find, without using a calculator, the value of sin2𝐴 given cos𝐴=1213 where 180𝐴<270.

  • A60169
  • B60169
  • C1013
  • D120169
  • E120169

Q6:

Find, without using a calculator, the value of sin𝜃2 given tan𝜃=158 where 3𝜋2<𝜃<2𝜋.

  • A53434
  • B2626
  • C33434
  • D41717

Q7:

𝐴𝐵𝐶 is a triangle where tan𝐶=815. Find the value of sin𝐴+𝐵2.

  • A534
  • B817
  • C417
  • D815

Q8:

Find the value of cos(𝜋+2𝐴) given sin(270+𝐴)=1517 where 3𝜋2<𝐴<2𝜋.

  • A161289
  • B161289
  • C240289
  • D240289

Q9:

Find, without using a calculator, 12𝑋1+2𝑋coscos given tan𝑋=4 where 𝑋𝜋,3𝜋2.

  • A116
  • B116
  • C16
  • D16

Q10:

Which of the following is equal to 12𝑥sin?

  • Asincos𝑥𝑥
  • Bcossin𝑥𝑥
  • C|𝑥𝑥|cossin
  • D|𝑥+𝑥|cossin
  • Ecossin𝑥+𝑥

Q11:

Use the addition formula to find an expression for sin2𝛼.

  • Asincossin2𝛼=𝛼+𝛼
  • Bsincossin2𝛼=2𝛼𝛼
  • Csincossin2𝛼=𝛼+𝛼
  • Dsincossin2𝛼=𝛼𝛼
  • Esincossin2𝛼=𝛼𝛼

Q12:

Find the value of 1+2𝐴1+2𝐴sincos given tan𝐴=526 where 0<𝐴<𝜋3 without using a calculator.

  • A9611,352
  • B1,352961
  • C9611,352
  • D1,352961

Q13:

Find the value of tancot15730+15730 and then tancot15730+15730 without using a calculator.

  • A22, 6
  • B2, 6
  • C22, 8
  • D22, 16

Q14:

Find the value of sin4𝑋 given 2𝑋𝑋2𝑋𝑋=926sincoscossin.

  • A913
  • B913
  • C926
  • D926

Q15:

Find, without using a calculator, the value of tan4𝑋 given sincos𝑋𝑋=14 where 𝑋𝜋2,3𝜋4.

  • A13
  • B13
  • C3
  • D3

Q16:

Use the addition formula to find an expression for cos2𝛼.

  • Acoscossin2𝛼=2𝛼+2𝛼
  • Bcoscossin2𝛼=𝛼𝛼
  • Ccoscossin2𝛼=𝛼+𝛼
  • Dcoscossin2𝛼=2𝛼2𝛼
  • Ecossincos2𝛼=2𝛼𝛼

Q17:

Which of the following is equal to 12𝑥cos?

  • A2|𝑥|sin
  • B|𝑥|sin
  • C2|𝑥|sin
  • D2|𝑥|cos
  • E2|𝑥|cos

Q18:

Using the half angle formulas, or otherwise, find the exact value of tan𝜋8.

  • A1+2
  • B1+2
  • C12
  • D2
  • E12

Q19:

Find the value of 11+tantan without using a calculator.

  • A12
  • B12
  • C12
  • D12

Q20:

Evaluate 3611230111230tantan without using a calculator.

  • A18
  • B36
  • C18
  • D1

Q21:

Simplify tantan3544141354414.

  • Atan3544142
  • Btan712828
  • Ctan7128282
  • Dtan7128282

Q22:

𝐴𝐵𝐶 is a triangle, where the ratio between its lengths 𝑎, 𝑏, and 𝑐 is 435. Find tan2𝐴.

  • A2425
  • B127
  • C247
  • D247
  • E127

Q23:

Find the value of tancot15730+15730 without using a calculator.

  • A2
  • B22
  • C22
  • D2

Q24:

Calculate cossinsincos151511515.

  • A12
  • B12
  • C32
  • D32

Q25:

Find, without using a calculator, the value of sincos2𝐵22𝐵 given cos𝐵=45 where 3𝜋2<𝐵<2𝜋.

  • A67
  • B625
  • C127
  • D1225

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