Worksheet: Solving a Trigonometric Equation by Factoring

In this worksheet, we will practice solving a trigonometric equation using factoring.

Q1:

Find the set of values satisfying 6 𝜃 7 𝜃 5 = 0 c o s c o s where 0 𝜃 < 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 }

Q2:

Find the set of values satisfying 5 𝜃 = 4 c o s where 0 𝜃 < 3 6 0 . Give the answer to the nearest minute.

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

Q3:

Find all the possible general solutions of 2 𝜃 2 𝜃 = 0 c o s c o s .

  • A 𝜋 2 + 𝑛 𝜋 , 𝜋 4 + 2 𝑛 𝜋 𝑛 : .
  • B 𝜋 2 + 𝑛 𝜋 , 𝜋 4 + 2 𝑛 𝜋 𝑛 : .
  • C 𝜋 2 + 𝑛 𝜋 , 𝜋 4 + 2 𝑛 𝜋 𝑛 : .
  • D 𝜋 2 + 𝑛 𝜋 , 𝜋 2 + 𝑛 𝜋 , 𝜋 4 + 2 𝑛 𝜋 , 𝜋 4 + 2 𝑛 𝜋 𝑛 : .
  • E 𝜋 2 + 2 𝑛 𝜋 , 𝜋 2 + 2 𝑛 𝜋 , 𝜋 4 + 2 𝑛 𝜋 , 𝜋 4 + 2 𝑛 𝜋 𝑛 : .

Q4:

Find the set of values satisfying 1 3 𝜃 7 6 𝜃 = 0 t a n t a n where 0 𝜃 < 3 6 0 . Give the answers to the nearest second.

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

Q5:

Find the set of values satisfying 2 𝜃 2 𝜃 2 = 0 s i n s i n given 1 8 0 𝜃 < 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 }

Q6:

Find the set of possible solutions of s i n c o s 𝜃 𝜃 = 0 given 𝜃 [ 0 , 3 6 0 [ .

  • A { 6 0 , 1 2 0 , 2 4 0 , 3 0 0 }
  • B { 3 0 , 1 5 0 , 2 1 0 , 3 3 0 }
  • C { 4 5 , 1 3 5 , 2 2 5 , 3 1 5 }

Q7:

Find the set of values satisfying t a n t a n 𝜃 + 𝜃 = 0 where 0 𝜃 < 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 }

Q8:

Find the set of values satisfying 2 2 𝜃 + 2 𝜃 = 0 c o s c o s given 0 < 𝜃 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 }

Q9:

Find all the possible general solutions of 2 𝜃 3 𝜃 = 0 c o s c o s .

  • A 𝜋 2 + 𝑛 𝜋 , 𝜋 6 + 2 𝑛 𝜋 𝑛 : .
  • B 𝜋 2 + 𝑛 𝜋 , 𝜋 6 + 2 𝑛 𝜋 𝑛 : .
  • C 𝜋 2 + 𝑛 𝜋 , 𝜋 6 + 2 𝑛 𝜋 𝑛 : .
  • D 𝜋 2 + 𝑛 𝜋 , 𝜋 2 + 𝑛 𝜋 , 𝜋 6 + 2 𝑛 𝜋 , 𝜋 6 + 2 𝑛 𝜋 𝑛 : .
  • E 𝜋 2 + 2 𝑛 𝜋 , 𝜋 2 + 2 𝑛 𝜋 , 𝜋 6 + 2 𝑛 𝜋 , 𝜋 6 + 2 𝑛 𝜋 𝑛 : .

Q10:

Find the set of values satisfying 7 1 𝜃 + 8 0 𝜃 = 0 t a n t a n where 0 𝜃 < 3 6 0 . Give the answers to the nearest second.

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

Q11:

Find the set of values satisfying 7 8 𝜃 + 4 9 𝜃 = 0 t a n t a n where 0 𝜃 < 3 6 0 . Give the answers to the nearest second.

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

Q12:

Find the set of values satisfying 6 𝜃 𝜃 1 = 0 c o s c o s where 0 𝜃 < 3 6 0 . Give the answers to the nearest minute.

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

Q13:

Consider the equation s i n c o s 𝜃 + 𝜃 = 2 , where 0 < 𝜃 3 6 0 . Call this Equation A.

Create Equation B by squaring both sides of Equation A. Use the fact that s i n c o s 𝜃 + 𝜃 = 1 to simplify Equation B.

  • A s i n c o s 𝜃 𝜃 = 2
  • B s i n c o s 𝜃 𝜃 = 1
  • C 2 𝜃 𝜃 = 1 s i n c o s
  • D 2 𝜃 𝜃 = 1 s i n c o s
  • E s i n c o s 𝜃 𝜃 = 1

Now, use a double angle formula to further simplify Equation B.

  • A s i n 2 𝜃 = 1
  • B c o s 𝜃 = 1
  • C c o s 2 𝜃 = 1
  • D s i n 𝜃 = 1
  • E s i n 2 𝜃 = 2

The solutions to Equation A are a subset of the solutions of Equation B. Using this, solve Equation A over the specified range.

  • A 𝜃 = 2 1 5
  • B 𝜃 = 3 0
  • C 𝜃 = 4 5
  • D 𝜃 = 6 0
  • E 𝜃 = 1 3 5

Q14:

Solve 2 𝜃 + 3 𝜃 = 2 s i n c o s , where 0 < 𝜃 2 𝜋 . Give your answer in radians to three significant figures.

  • A 𝜃 = 1 . 6 9 , 2 . 1 4
  • B 𝜃 = 0 . 4 7 1 , 2 . 1 7
  • C 𝜃 = 0 . 2 4 1 , 1 . 8 6
  • D 𝜃 = 0 . 2 2 1 , 1 . 1 5
  • E 𝜃 = 0 . 3 9 6 , 2 . 9 5

Q15:

If 𝜃 [ 0 , 1 8 0 [ and s i n c o s 𝜃 + 𝜃 = 1 , find the possible values of 𝜃 .

  • A 9 0 , 1 8 0
  • B 0 , 1 8 0
  • C 4 5 , 9 0
  • D 0 , 9 0
  • E 0 , 4 5

Q16:

By first squaring both sides, or otherwise, solve the equation 4 𝜃 4 𝜃 = 3 s i n c o s , where 0 < 𝜃 3 6 0 . Be careful to remove any extraneous solutions. Give your answers to two decimal places.

  • A 𝜃 = 6 5 . 1 8 , 2 0 5 . 1 4
  • B 𝜃 = 8 6 . 1 4 , 2 1 2 . 5 7
  • C 𝜃 = 4 7 . 3 5 , 1 9 5 . 1 2
  • D 𝜃 = 6 2 . 8 3 , 2 0 7 . 1 7
  • E 𝜃 = 7 7 . 2 4 , 2 1 0 . 5 7

Q17:

Find the set of values satisfying 9 7 𝜃 + 6 0 𝜃 = 0 s i n c o s where 0 < 𝜃 < 3 6 0 . Give the answers to the nearest second.

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

Q18:

If 𝜃 ( 1 8 0 , 3 6 0 ) and s i n c o s 𝜃 + 𝜃 = 1 , find the value of 𝜃 .

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