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Worksheet: Solving a Trigonometric Equation by Quadratic Methods

Q1:

Find all the possible general solutions of 2 πœƒ βˆ’ √ 2 πœƒ = 0 c o s c o s 2 .

  • A πœ‹ 2 + 𝑛 πœ‹ , πœ‹ 4 + 2 𝑛 πœ‹ 𝑛 ∈ β„€ : .
  • B πœ‹ 2 + 𝑛 πœ‹ , βˆ’ πœ‹ 4 + 2 𝑛 πœ‹ 𝑛 ∈ β„€ : .
  • C βˆ’ πœ‹ 2 + 𝑛 πœ‹ , βˆ’ πœ‹ 4 + 2 𝑛 πœ‹ 𝑛 ∈ β„€ : .
  • D πœ‹ 2 + 𝑛 πœ‹ , βˆ’ πœ‹ 2 + 𝑛 πœ‹ , πœ‹ 4 + 2 𝑛 πœ‹ , βˆ’ πœ‹ 4 + 2 𝑛 πœ‹ 𝑛 ∈ β„€ : .
  • E πœ‹ 2 + 2 𝑛 πœ‹ , βˆ’ πœ‹ 2 + 2 𝑛 πœ‹ , πœ‹ 4 + 2 𝑛 πœ‹ , βˆ’ πœ‹ 4 + 2 𝑛 πœ‹ 𝑛 ∈ β„€ : .

Q2:

Find all the possible general solutions of 2 πœƒ βˆ’ √ 3 πœƒ = 0 c o s c o s 2 .

  • A πœ‹ 2 + 𝑛 πœ‹ , πœ‹ 6 + 2 𝑛 πœ‹ 𝑛 ∈ β„€ : .
  • B πœ‹ 2 + 𝑛 πœ‹ , βˆ’ πœ‹ 6 + 2 𝑛 πœ‹ 𝑛 ∈ β„€ : .
  • C βˆ’ πœ‹ 2 + 𝑛 πœ‹ , βˆ’ πœ‹ 6 + 2 𝑛 πœ‹ 𝑛 ∈ β„€ : .
  • D πœ‹ 2 + 𝑛 πœ‹ , βˆ’ πœ‹ 2 + 𝑛 πœ‹ , πœ‹ 6 + 2 𝑛 πœ‹ , βˆ’ πœ‹ 6 + 2 𝑛 πœ‹ 𝑛 ∈ β„€ : .
  • E πœ‹ 2 + 2 𝑛 πœ‹ , βˆ’ πœ‹ 2 + 2 𝑛 πœ‹ , πœ‹ 6 + 2 𝑛 πœ‹ , βˆ’ πœ‹ 6 + 2 𝑛 πœ‹ 𝑛 ∈ β„€ : .

Q3:

Find the set of values satisfying 1 3 πœƒ βˆ’ 7 6 πœƒ = 0 t a n t a n 2 where 0 ≀ πœƒ < 3 6 0 ∘ ∘ . Give the answers to the nearest second.

  • A { 8 0 1 7 β€² 3 6 β€² β€² , 2 6 0 1 7 β€² 3 6 β€² β€² } ∘ ∘
  • B { 0 0 β€² 0 β€² β€² , 1 8 0 0 β€² 0 β€² β€² , 8 0 1 7 β€² 3 6 β€² β€² , 9 9 4 2 β€² 2 4 β€² β€² } ∘ ∘ ∘ ∘
  • C { 0 0 β€² 0 β€² β€² , 1 8 0 0 β€² 0 β€² β€² , 3 6 0 0 β€² 0 β€² β€² , 8 0 1 7 β€² 3 6 β€² β€² , 2 6 0 1 7 β€² 3 6 β€² β€² } ∘ ∘ ∘ ∘ ∘
  • D { 0 0 β€² 0 β€² β€² , 1 8 0 0 β€² 0 β€² β€² , 8 0 1 7 β€² 3 6 β€² β€² , 2 6 0 1 7 β€² 3 6 β€² β€² } ∘ ∘ ∘ ∘
  • E { 0 0 β€² 0 β€² β€² , 1 8 0 0 β€² 0 β€² β€² , 2 6 0 1 7 β€² 3 6 β€² β€² , 9 9 4 2 β€² 2 4 β€² β€² } ∘ ∘ ∘ ∘

Q4:

Find the set of values satisfying 7 1 πœƒ + 8 0 πœƒ = 0 t a n t a n 2 where 0 ≀ πœƒ < 3 6 0 ∘ ∘ . Give the answers to the nearest second.

  • A { 1 3 1 3 5 β€² 2 1 β€² β€² , 3 1 1 3 5 β€² 2 1 β€² β€² } ∘ ∘
  • B { 0 0 β€² 0 β€² β€² , 1 8 0 0 β€² 0 β€² β€² , 4 8 2 4 β€² 3 9 β€² β€² , 3 1 1 3 5 β€² 2 1 β€² β€² } ∘ ∘ ∘ ∘
  • C { 4 8 2 4 β€² 3 9 β€² β€² , 2 2 8 2 4 β€² 3 9 β€² β€² } ∘ ∘
  • D { 0 0 β€² 0 β€² β€² , 1 8 0 0 β€² 0 β€² β€² , 1 3 1 3 5 β€² 2 1 β€² β€² , 3 1 1 3 5 β€² 2 1 β€² β€² } ∘ ∘ ∘ ∘
  • E { 4 8 2 4 β€² 3 9 β€² β€² , 1 3 1 3 5 β€² 2 1 β€² β€² } ∘ ∘

Q5:

Find the set of values satisfying 7 8 πœƒ + 4 9 πœƒ = 0 t a n t a n 2 where 0 ≀ πœƒ < 3 6 0 ∘ ∘ . Give the answers to the nearest second.

  • A { 1 4 7 5 1 β€² 4 6 β€² β€² , 3 2 7 5 1 β€² 4 6 β€² β€² } ∘ ∘
  • B { 0 0 β€² 0 β€² β€² , 1 8 0 0 β€² 0 β€² β€² , 3 2 8 β€² 1 4 β€² β€² , 3 2 7 5 1 β€² 4 6 β€² β€² } ∘ ∘ ∘ ∘
  • C { 3 2 8 β€² 1 4 β€² β€² , 2 1 2 8 β€² 1 4 β€² β€² } ∘ ∘
  • D { 0 0 β€² 0 β€² β€² , 1 8 0 0 β€² 0 β€² β€² , 1 4 7 5 1 β€² 4 6 β€² β€² , 3 2 7 5 1 β€² 4 6 β€² β€² } ∘ ∘ ∘ ∘
  • E { 3 2 8 β€² 1 4 β€² β€² , 1 4 7 5 1 β€² 4 6 β€² β€² } ∘ ∘

Q6:

Find the set of values satisfying 2 πœƒ βˆ’ √ 2 πœƒ βˆ’ 2 = 0 s i n s i n 2 given 1 8 0 ≀ πœƒ < 3 6 0 ∘ ∘ .

  • A { 1 3 5 , 2 2 5 } ∘ ∘
  • B { 4 5 , 1 3 5 } ∘ ∘
  • C { 1 3 5 , 3 1 5 } ∘ ∘
  • D { 2 2 5 , 3 1 5 } ∘ ∘

Q7:

Find the set of values satisfying 2 √ 2 πœƒ + 2 πœƒ = 0 c o s c o s 2 given 0 < πœƒ ≀ 3 6 0 ∘ ∘ .

  • A { 0 , 1 3 5 , 1 8 0 , 2 2 5 } ∘ ∘ ∘ ∘
  • B { 4 5 , 9 0 , 2 7 0 , 3 1 5 } ∘ ∘ ∘ ∘
  • C { 0 , 4 5 , 1 3 5 , 1 8 0 } ∘ ∘ ∘ ∘
  • D { 9 0 , 1 3 5 , 2 2 5 , 2 7 0 } ∘ ∘ ∘ ∘

Q8:

Find the set of values satisfying 6 πœƒ βˆ’ 7 πœƒ βˆ’ 5 = 0 c o s c o s 2 where 0 ≀ πœƒ < 3 6 0 ∘ ∘ . Give the answers to the nearest minute.

  • A { 6 0 , 3 0 0 } ∘ ∘
  • B { 6 0 , 2 4 0 } ∘ ∘
  • C { 1 2 0 , 3 0 0 } ∘ ∘
  • D { 1 2 0 , 2 4 0 } ∘ ∘

Q9:

Find the set of values satisfying 6 πœƒ βˆ’ πœƒ βˆ’ 1 = 0 c o s c o s 2 where 0 ≀ πœƒ < 3 6 0 ∘ ∘ . Give the answers to the nearest minute.

  • A { 1 2 0 , 2 4 0 , 7 0 3 2 β€² , 2 8 9 2 8 β€² } ∘ ∘ ∘ ∘
  • B { 6 0 , 3 0 0 , 7 0 3 2 β€² , 2 5 0 3 2 β€² } ∘ ∘ ∘ ∘
  • C { 1 2 0 , 3 0 0 , 1 0 9 2 8 β€² , 2 8 9 2 8 β€² } ∘ ∘ ∘ ∘
  • D { 6 0 , 1 0 9 2 8 β€² , 3 0 0 , 2 5 0 3 2 β€² } ∘ ∘ ∘ ∘

Q10:

Find the set of possible solutions of given .

  • A
  • B
  • C

Q11:

Find the set of values satisfying 5 πœƒ = 4 c o s 2 where 0 ≀ πœƒ < 3 6 0 ∘ ∘ . Give the answer to the nearest minute.

  • A { 3 6 3 4 β€² , 1 4 3 2 6 β€² , 2 1 6 3 4 β€² , 3 2 3 2 6 β€² } ∘ ∘ ∘ ∘
  • B { 6 3 2 6 β€² , 1 1 6 3 4 β€² , 2 4 3 2 6 β€² , 2 9 6 3 4 β€² } ∘ ∘ ∘ ∘
  • C { 7 3 2 6 β€² , 1 0 6 3 4 β€² , 2 5 3 2 6 β€² , 2 8 6 3 4 β€² } ∘ ∘ ∘ ∘
  • D { 2 6 3 4 β€² , 1 5 3 2 6 β€² , 2 0 6 3 4 β€² , 3 3 3 2 6 β€² } ∘ ∘ ∘ ∘

Q12:

Find the set of values satisfying t a n t a n 2 πœƒ + πœƒ = 0 where 0 ≀ πœƒ < 1 8 0 ∘ ∘ .

  • A { 4 5 , 1 3 5 , 0 , 9 0 } ∘ ∘ ∘ ∘
  • B { 1 3 5 , 4 5 , 9 0 , 2 7 0 } ∘ ∘ ∘ ∘
  • C { 1 3 5 , 2 2 5 , 0 , 1 8 0 } ∘ ∘ ∘ ∘
  • D { 1 3 5 , 3 1 5 , 0 , 1 8 0 } ∘ ∘ ∘ ∘