Worksheet: Double-Angle and Half-Angle Identities

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

Q1:

Find the value of c o s 2 𝐴 given c o s 𝐴 = − 3 5 where 9 0 < 𝐴 < 1 8 0 ∘ ∘ without using a calculator.

  • A 2 4 2 5
  • B − 2 4 2 5
  • C 7 2 5
  • D − 7 2 5

Q2:

Find the value of c o s 𝐴 2 given s i n 𝐴 2 = 3 5 without using a calculator.

  • A 3 4
  • B − 4 5
  • C − 3 4
  • D 4 5

Q3:

Find, without using a calculator, the value of s i n 2 𝐴 given t a n 𝐴 = − 5 1 2 where 3 𝜋 2 < 𝐴 < 2 𝜋 .

  • A 1 1 9 1 6 9
  • B − 6 0 1 6 9
  • C − 1 1 9 1 6 9
  • D − 1 2 0 1 6 9

Q4:

Find the value of c o s 𝜃 2 given c o s 𝜃 = 1 5 1 7 where 0 < 𝜃 < 9 0 ∘ ∘ without using a calculator.

  • A 3 √ 1 0 1 0
  • B √ 1 7 1 7
  • C 3 √ 5 5
  • D 4 √ 1 7 1 7

Q5:

Knowing that 5 𝑥 + 1 2 𝑥 = 1 3 s i n c o s , find s i n 𝑥 and c o s 𝑥 .

  • A s i n c o s 𝑥 = 1 3 5 , 𝑥 = 1 3 1 2
  • B s i n c o s 𝑥 = 1 2 1 3 , 𝑥 = 5 1 3
  • C s i n c o s 𝑥 = 1 3 1 2 , 𝑥 = 1 3 5
  • D s i n c o s 𝑥 = 5 1 3 , 𝑥 = 1 2 1 3
  • E s i n c o s 𝑥 = − 5 1 3 , 𝑥 = − 1 2 1 3

Q6:

Knowing that 3 𝑥 − 4 𝑥 = 5 s i n c o s , find s i n 𝑥 and c o s 𝑥 .

  • A s i n c o s 𝑥 = 3 5 , 𝑥 = 4 5
  • B s i n c o s 𝑥 = − 3 5 , 𝑥 = 4 5
  • C s i n c o s 𝑥 = 5 3 , 𝑥 = − 5 4
  • D s i n c o s 𝑥 = 3 5 , 𝑥 = − 4 5
  • E s i n c o s 𝑥 = − 5 3 , 𝑥 = 5 4

Q7:

Find, without using a calculator, the value of s i n 2 𝐴 given c o s 𝐴 = − 1 2 1 3 where 1 8 0 ≤ 𝐴 < 2 7 0 ∘ ∘ .

  • A 1 0 1 3
  • B 6 0 1 6 9
  • C − 1 2 0 1 6 9
  • D 1 2 0 1 6 9
  • E − 6 0 1 6 9

Q8:

Find, without using a calculator, the value of s i n 𝜃 2 given t a n 𝜃 = − 1 5 8 where 3 𝜋 2 < 𝜃 < 2 𝜋 .

  • A 4 √ 1 7 1 7
  • B 5 √ 3 4 3 4
  • C √ 2 6 2 6
  • D 3 √ 3 4 3 4

Q9:

𝐴 𝐵 𝐶 is a triangle where t a n 𝐶 = 8 1 5 . Find the value of s i n 𝐴 + 𝐵 2 .

  • A 5 √ 3 4
  • B 8 √ 1 7
  • C 8 √ 1 5
  • D 4 √ 1 7

Q10:

Find the value of c o s ( 𝜋 + 2 𝐴 ) given s i n ( 2 7 0 + 𝐴 ) = − 1 5 1 7 ∘ where 3 𝜋 2 < 𝐴 < 2 𝜋 .

  • A 2 4 0 2 8 9
  • B − 2 4 0 2 8 9
  • C 1 6 1 2 8 9
  • D − 1 6 1 2 8 9

Q11:

Find, without using a calculator, 1 − 2 𝑋 1 + 2 𝑋 c o s c o s given t a n 𝑋 = 4 where 𝑋 ∈ 𝜋 , 3 𝜋 2 .

  • A − 1 6
  • B 1 1 6
  • C − 1 1 6
  • D16

Q12:

Which of the following is equal to √ 1 − 2 𝑥 s i n ?

  • A | 𝑥 + 𝑥 | c o s s i n
  • B c o s s i n 𝑥 − 𝑥
  • C c o s s i n 𝑥 + 𝑥
  • D | 𝑥 − 𝑥 | c o s s i n
  • E s i n c o s 𝑥 − 𝑥

Q13:

Use the addition formula to find an expression for s i n 2 𝛼 .

  • A s i n c o s s i n 2 𝛼 = 𝛼 + 𝛼
  • B s i n c o s s i n 2 𝛼 = 𝛼 𝛼
  • C s i n c o s s i n 2 𝛼 = 𝛼 − 𝛼 2 2
  • D s i n c o s s i n 2 𝛼 = 2 𝛼 𝛼
  • E s i n c o s s i n 2 𝛼 = 𝛼 + 𝛼 2 2

Q14:

Given that s i n c o s 𝑋 + 𝑋 = − 7 1 3 and 𝜋 < 𝑋 < 3 𝜋 2 , determine the possible values of c o s 2 𝑋 .

  • A 1 6 9 1 1 9 , − 1 6 9 1 1 9
  • B 1 2 0 1 6 9 , − 1 2 0 1 6 9
  • C 1 6 9 1 2 0 , − 1 6 9 1 2 0
  • D 1 1 9 1 6 9 , − 1 1 9 1 6 9

Q15:

Find the value of 1 + 2 𝐴 1 + 2 𝐴 s i n c o s given t a n 𝐴 = 5 2 6 where 0 < 𝐴 < 𝜋 3 without using a calculator.

  • A − 9 6 1 1 3 5 2
  • B 1 3 5 2 9 6 1
  • C − 1 3 5 2 9 6 1
  • D 9 6 1 1 3 5 2

Q16:

Find the value of t a n c o t 1 5 7 3 0 ′ + 1 5 7 3 0 ′ ∘ ∘ and then t a n c o t 2 ∘ 2 ∘ 1 5 7 3 0 ′ + 1 5 7 3 0 ′ without using a calculator.

  • A √ 2 , 6
  • B − 2 √ 2 , 8
  • C 2 √ 2 , 16
  • D − 2 √ 2 , 6

Q17:

Find the value of s i n 4 𝑋 given 2 𝑋 𝑋 − 2 𝑋 𝑋 = 9 2 6 s i n c o s c o s s i n 3 3 .

  • A − 9 1 3
  • B 9 2 6
  • C − 9 2 6
  • D 9 1 3

Q18:

Find, without using a calculator, the value of t a n 4 𝑋 given s i n c o s 𝑋 𝑋 = − 1 4 where 𝑋 ∈ 𝜋 2 , 3 𝜋 4 .

  • A − √ 3
  • B 1 √ 3
  • C − 1 √ 3
  • D √ 3

Q19:

Use the addition formula to find an expression for c o s 2 𝛼 .

  • A c o s c o s s i n 2 𝛼 = 2 𝛼 − 2 𝛼
  • B c o s c o s s i n 2 𝛼 = 𝛼 + 𝛼 2 2
  • C c o s c o s s i n 2 𝛼 = 2 𝛼 + 2 𝛼
  • D c o s c o s s i n 2 𝛼 = 𝛼 − 𝛼 2 2
  • E c o s s i n c o s 2 𝛼 = 2 𝛼 𝛼

Q20:

Which of the following is equal to √ 1 − 2 𝑥 c o s ?

  • A √ 2 | 𝑥 | c o s
  • B 2 | 𝑥 | s i n
  • C 2 | 𝑥 | c o s
  • D √ 2 | 𝑥 | s i n
  • E | 𝑥 | s i n

Q21:

Using the half angle formulas, or otherwise, find the exact value of t a n 𝜋 8 .

  • A 1 + √ 2
  • B − 1 − √ 2
  • C √ 2
  • D − 1 + √ 2
  • E 1 2

Q22:

Find the value of 1 − 1 + t a n t a n 2 7 𝜋 8 2 7 𝜋 8 without using a calculator.

  • A 1 2
  • B − 1 √ 2
  • C − 1 2
  • D 1 √ 2

Q23:

Evaluate 3 6 1 1 2 3 0 ′ 1 − 1 1 2 3 0 ′ t a n t a n ∘ 2 ∘ without using a calculator.

  • A1
  • B − 1 8
  • C36
  • D18

Q24:

Simplify t a n t a n 3 5 4 4 ′ 1 4 ′ ′ 1 − 3 5 4 4 ′ 1 4 ′ ′ ∘ 2 ∘ .

  • A t a n 2 ∘ 7 1 2 8 ′ 2 8 ′ ′ 2
  • B t a n 3 5 4 4 ′ 1 4 ′ ′ 2 ∘
  • C t a n 7 1 2 8 ′ 2 8 ′ ′ ∘
  • D t a n 7 1 2 8 ′ 2 8 ′ ′ 2 ∘

Q25:

𝐴 𝐵 𝐶 is a triangle, where the ratio between its lengths 𝑎 , 𝑏 , and 𝑐 is 4 ∶ 3 ∶ 5 . Find t a n 2 𝐴 .

  • A − 1 2 7
  • B 2 4 7
  • C 2 4 2 5
  • D − 2 4 7
  • E 1 2 7

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