Worksheet: Evaluating Trigonometric Expressions Using Double-Angle Identities

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In this worksheet, we will practice finding the function values of 2θ given information about θ, and vice versa.

Q1:

Find the value of c o s 1 5 ∘ without using a calculator.

  • A √ 6 + √ 3 2
  • B √ 6 − √ 2 4
  • C √ 6 − √ 3 2
  • D √ 6 + √ 2 4

Q2:

Find the value of 2 1 5 7 3 0 ′ − 1 c o s 2 ∘ without using a calculator.

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

Q3:

Find the value of t a n 2 2 3 0 ′ ∘ without using a calculator.

  • A 1 − √ 2
  • B 2 + √ 3
  • C 2 − √ 3
  • D − 1 + √ 2

Q4:

Find, without using a calculator, t a n 2 𝐴 given c o s 𝐴 = 9 1 5 where 0 < 𝐴 < 9 0 ∘ ∘ .

  • A − 8 2 1
  • B − 1 2 7
  • C − 2 1 6 7
  • D − 2 4 7

Q5:

Find, without using a calculator, the value of t a n 2 𝐴 given s i n 𝐴 = 2 4 2 5 , where 0 < 𝐴 < 9 0 ∘ ∘ .

  • A − 4 8 5 2 7
  • B − 1 6 8 5 2 7
  • C − 2 3 5 2 5 2 7
  • D − 3 3 6 5 2 7

Q6:

Find the value of 2 8 1 1 1 5 ′ 1 1 1 5 ′ 2 2 3 0 ′ s i n c o s c o s ∘ ∘ ∘ .

  • A − 7 √ 2
  • B 1 7 √ 2
  • C − 1 7 √ 2
  • D 7 √ 2

Q7:

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

  • A 4 1 8 4 1
  • B 4 2 0 8 4 1
  • C 4 0 0 8 4 1
  • D 8 4 0 8 4 1

Q8:

Evaluate c o s s i n 2 ∘ 2 ∘ 1 6 5 − 1 6 5 without using a calculator.

  • A − 1 4
  • B √ 3 4
  • C − √ 3 2
  • D √ 3 2

Q9:

Find the value of c o s ( 2 𝐴 + 9 0 ) ∘ given 5 𝐴 = − 1 2 t a n where 2 7 0 < 𝐴 < 3 6 0 ∘ ∘ .

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

Q10:

Find the value of s i n 𝜃 2 given s i n 𝜃 = − 8 1 7 where 1 8 0 < 𝜃 < 2 7 0 ∘ ∘ .

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

Q11:

Find the value of s i n 2 𝐴 given c o s 2 𝐴 = 9 2 5 where 𝐴 ∈ 𝜋 , 3 𝜋 2 .

  • A 8 5
  • B 1 2 2 5
  • C − 2 4 2 5
  • D 2 4 2 5
  • E − 1 2 2 5

Q12:

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

  • A √ 1 0 1 0
  • B √ 2 6 2 6
  • C √ 5 5
  • D 5 √ 2 6 2 6

Q13:

Find the value of s i n c o s 𝑋 + 𝑋 given s i n 2 𝑋 = − 1 2 1 3 where 2 𝑋 ∈ ] 0 , 2 𝜋 [ .

  • A 2 √ 1 3
  • B 1 1 3
  • C 1 + √ 1 3 √ 1 3
  • D 1 √ 1 3
  • E √ 1 3

Q14:

Find the value of c o s s i n 𝐴 + 2 𝐵 given t a n 𝐴 2 = 3 2 where 𝐴 ∈ 𝜋 2 , 𝜋 and t a n 𝐵 = 3 5 where 𝐵 ∈ 0 , 𝜋 2 .

  • A 2 1 7
  • B − 2 1 7
  • C − 1 1 0 2 2 1
  • D 1 1 0 2 2 1
  • E − 1 0 0 2 2 1

Q15:

Find, without using a calculator, the value of s i n 2 𝐴 given c s c 𝐴 = − 5 3 where 2 7 0 < 𝐴 < 3 6 0 ∘ ∘ .

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

Q16:

Find s i n 2 𝑎 given c o s s i n 𝑎 − 𝑎 = 1 1 2 .

  • A 1 1 4 4
  • B 1 1 1 2
  • C 1 2 8 8
  • D 1 4 3 1 4 4

Q17:

Find the value of s i n 2 𝑋 given s i n c o s 𝑋 + 𝑋 = − 1 √ 3 .

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

Q18:

Find the value of c o s 𝐴 2 given c o s 2 𝐴 = 1 4 where 𝐴 ∈ 𝜋 , 3 𝜋 2 .

  • A 2 √ 5 5
  • B − 3 4
  • C 3 √ 1 0 1 0
  • D − √ 4 4

Q19:

Find the value of c o s 𝜃 2 given s i n 𝜃 = 3 5 where 0 < 𝜃 < 9 0 ∘ ∘ .

  • A 5 √ 3 4 3 4
  • B √ 1 0 1 0
  • C 2 √ 5 5
  • D 3 √ 1 0 1 0

Q20:

Find the value of c o s 𝐴 given c o s ( 2 𝐴 ) = − 7 2 5 where 𝐴 is a positive acute angle.

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

Q21:

Find, without using a calculator, the value of c o s 2 𝐴 given t a n 𝐴 = 3 4 where 𝜋 < 𝐴 < 3 𝜋 2 .

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

Q22:

Find, without using a calculator, the value of c o s 𝜃 2 given t a n 𝜃 = 4 3 where 𝜋 < 𝜃 < 3 𝜋 2 .

  • A 2 √ 5 5
  • B − 2 √ 5 5
  • C √ 5 5
  • D − √ 5 5

Q23:

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

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

Q24:

Evaluate c o s s i n 4 ∘ 4 ∘ 1 1 2 3 0 ′ − 1 1 2 3 0 ′ without using a calculator.

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

Q25:

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

  • A 6 7
  • B 1 2 7
  • C 9 6 7
  • D 2 4 7

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