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Lesson: Double-Angle and Half-Angle Identities

In this lesson, we will learn how to use the Pythagorean identity and double-angle formulas to evaluate trigonometric values.

Sample Question Videos

02:23
01:38

Worksheet • 25 Questions • 2 Videos

Q1:

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

  • A βˆ’ 7 2 5
  • B βˆ’ 2 4 2 5
  • C 7 2 5
  • D 2 4 2 5

Q2:

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

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

Q3:

Find, without using a calculator, the value of s i n 2 𝐴 given t a n 𝐴 = βˆ’ 5 1 2 where 3 πœ‹ 2 < 𝐴 < 2 πœ‹ .

  • A βˆ’ 1 2 0 1 6 9
  • B βˆ’ 6 0 1 6 9
  • C βˆ’ 1 1 9 1 6 9
  • D 1 1 9 1 6 9

Q4:

Find the value of c o s ο€½ πœƒ 2  given c o s πœƒ = 1 5 1 7 where 0 < πœƒ < 9 0 ∘ ∘ without using a calculator.

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

Q5:

Knowing that 5 π‘₯ + 1 2 π‘₯ = 1 3 s i n c o s , find s i n π‘₯ and c o s π‘₯ .

  • A s i n c o s π‘₯ = 5 1 3 , π‘₯ = 1 2 1 3
  • B s i n c o s π‘₯ = βˆ’ 5 1 3 , π‘₯ = βˆ’ 1 2 1 3
  • C s i n c o s π‘₯ = 1 2 1 3 , π‘₯ = 5 1 3
  • D s i n c o s π‘₯ = 1 3 1 2 , π‘₯ = 1 3 5
  • E s i n c o s π‘₯ = 1 3 5 , π‘₯ = 1 3 1 2

Q6:

Knowing that 3 π‘₯ βˆ’ 4 π‘₯ = 5 s i n c o s , find s i n π‘₯ and c o s π‘₯ .

  • A s i n c o s π‘₯ = 3 5 , π‘₯ = βˆ’ 4 5
  • B s i n c o s π‘₯ = βˆ’ 5 3 , π‘₯ = 5 4
  • C s i n c o s π‘₯ = βˆ’ 3 5 , π‘₯ = 4 5
  • D s i n c o s π‘₯ = 5 3 , π‘₯ = βˆ’ 5 4
  • E s i n c o s π‘₯ = 3 5 , π‘₯ = 4 5

Q7:

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

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

Q8:

Find, without using a calculator, the value of s i n ο€½ πœƒ 2  given t a n πœƒ = βˆ’ 1 5 8 where 3 πœ‹ 2 < πœƒ < 2 πœ‹ .

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

Q9:

𝐴 𝐡 𝐢 is a triangle where t a n 𝐢 = 8 1 5 . Find the value of s i n ο€Ό 𝐴 + 𝐡 2  .

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

Q10:

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

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

Q11:

Find, without using a calculator, 1 βˆ’ 2 𝑋 1 + 2 𝑋 c o s c o s given t a n 𝑋 = 4 where 𝑋 ∈  πœ‹ , 3 πœ‹ 2  .

  • A16
  • B 1 1 6
  • C βˆ’ 1 1 6
  • D βˆ’ 1 6

Q12:

Which of the following is equal to √ 1 βˆ’ 2 π‘₯ s i n ?

  • A | π‘₯ βˆ’ π‘₯ | c o s s i n
  • B s i n c o s π‘₯ βˆ’ π‘₯
  • C c o s s i n π‘₯ βˆ’ π‘₯
  • D c o s s i n π‘₯ + π‘₯
  • E | π‘₯ + π‘₯ | c o s s i n

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 = 2 𝛼 𝛼
  • B s i n c o s s i n 2 𝛼 = 𝛼 + 𝛼 2 2
  • C s i n c o s s i n 2 𝛼 = 𝛼 𝛼
  • D s i n c o s s i n 2 𝛼 = 𝛼 βˆ’ 𝛼 2 2
  • E s i n c o s s i n 2 𝛼 = 𝛼 + 𝛼

Q14:

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

  • A 1 1 9 1 6 9 , βˆ’ 1 1 9 1 6 9
  • B 1 2 0 1 6 9 , βˆ’ 1 2 0 1 6 9
  • C 1 6 9 1 2 0 , βˆ’ 1 6 9 1 2 0
  • D 1 6 9 1 1 9 , βˆ’ 1 6 9 1 1 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œ‹ 3 without using a calculator.

  • A 9 6 1 1 3 5 2
  • B 1 3 5 2 9 6 1
  • C βˆ’ 1 3 5 2 9 6 1
  • D βˆ’ 9 6 1 1 3 5 2

Q16:

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

  • A βˆ’ 2 √ 2 , 6
  • B βˆ’ 2 √ 2 , 8
  • C 2 √ 2 , 16
  • D √ 2 , 6

Q17:

Find the value of s i n 4 𝑋 given 2 𝑋 𝑋 βˆ’ 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 βˆ’ 9 2 6
  • D βˆ’ 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 = βˆ’ 1 4 where 𝑋 ∈  πœ‹ 2 , 3 πœ‹ 4  .

  • A √ 3
  • B 1 √ 3
  • C βˆ’ 1 √ 3
  • D βˆ’ √ 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 = 𝛼 βˆ’ 𝛼 2 2
  • B c o s s i n c o s 2 𝛼 = 2 𝛼 𝛼
  • C c o s c o s s i n 2 𝛼 = 𝛼 + 𝛼 2 2
  • D c o s c o s s i n 2 𝛼 = 2 𝛼 + 2 𝛼
  • E c o s c o s s i n 2 𝛼 = 2 𝛼 βˆ’ 2 𝛼

Q20:

Which of the following is equal to √ 1 βˆ’ 2 π‘₯ c o s ?

  • A √ 2 | π‘₯ | s i n
  • B | π‘₯ | s i n
  • C 2 | π‘₯ | s i n
  • D 2 | π‘₯ | c o s
  • E √ 2 | π‘₯ | c o s

Q21:

Using the half angle formulas, or otherwise, find the exact value of t a n ο€» πœ‹ 8  .

  • A βˆ’ 1 + √ 2
  • B 1 2
  • C βˆ’ 1 βˆ’ √ 2
  • D √ 2
  • E 1 + √ 2

Q22:

Find the value of 1 βˆ’ ο€»  1 + ο€»  t a n t a n 2 7 πœ‹ 8 2 7 πœ‹ 8 without using a calculator.

  • A 1 √ 2
  • B βˆ’ 1 √ 2
  • C βˆ’ 1 2
  • D 1 2

Q23:

Evaluate 3 6 1 1 2 3 0 β€² 1 βˆ’ 1 1 2 3 0 β€² t a n t a n ∘ 2 ∘ without using a calculator.

  • A18
  • B βˆ’ 1 8
  • C36
  • D1

Q24:

Simplify t a n t a n 3 5 4 4 β€² 1 4 β€² β€² 1 βˆ’ 3 5 4 4 β€² 1 4 β€² β€² ∘ 2 ∘ .

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

Q25:

𝐴 𝐡 𝐢 is a triangle, where the ratio between its lengths π‘Ž , 𝑏 , and 𝑐 is 4 ∢ 3 ∢ 5 . Find t a n 2 𝐴 .

  • A βˆ’ 2 4 7
  • B 1 2 7
  • C 2 4 7
  • D 2 4 2 5
  • E βˆ’ 1 2 7
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