Worksheet: Solving a Trigonometric Equation

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

Q1:

Find the set of values satisfying 6𝜃7𝜃5=0coscos where 0𝜃<360. Give the answers to the nearest minute.

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

Q2:

Find the set of values satisfying 5𝜃=4cos where 0𝜃<360. Give the answer to the nearest minute.

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

Q3:

Find all the possible general solutions of 2𝜃2𝜃=0coscos.

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

Q4:

Find the set of values satisfying 13𝜃76𝜃=0tantan where 0𝜃<360. Give the answers to the nearest second.

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

Q5:

Find the set of values satisfying 2𝜃2𝜃2=0sinsin given 180𝜃<360.

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

Q6:

Find the set of possible solutions of sincos𝜃𝜃=0 given 𝜃[0,360).

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

Q7:

Find the set of values satisfying tantan𝜃+𝜃=0 where 0𝜃<180.

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

Q8:

Find the set of values satisfying 22𝜃+2𝜃=0coscos given 0<𝜃360.

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

Q9:

Find all the possible general solutions of 2𝜃3𝜃=0coscos.

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

Q10:

Find the set of values satisfying 71𝜃+80𝜃=0tantan where 0𝜃<360. Give the answers to the nearest second.

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

Q11:

Find the set of values satisfying 78𝜃+49𝜃=0tantan where 0𝜃<360. Give the answers to the nearest second.

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

Q12:

Find the set of values satisfying 6𝜃𝜃1=0coscos where 0𝜃<360. Give the answers to the nearest minute.

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

Q13:

Consider the equation sincos𝜃+𝜃=2, where 0<𝜃360. Call this Equation A.

Create Equation B by squaring both sides of Equation A. Use the fact that sincos𝜃+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 s i n c o s 𝜃 𝜃 = 1
  • D 2 𝜃 𝜃 = 1 s i n c o s
  • E 2 𝜃 𝜃 = 1 s i n c o s

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

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

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

Q14:

Solve 2𝜃+3𝜃=2sincos, where 0<𝜃2𝜋. Give your answer in radians to three significant figures.

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

Q15:

If 𝜃[0,180) and sincos𝜃+𝜃=1, find the possible values of 𝜃.

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

Q16:

By first squaring both sides, or otherwise, solve the equation 4𝜃4𝜃=3sincos, where 0<𝜃360. Be careful to remove any extraneous solutions. Give your answers to two decimal places.

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

Q17:

Find the set of values satisfying 97𝜃+60𝜃=0sincos where 0<𝜃<360. Give the answers to the nearest second.

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

Q18:

If 𝜃(180,360) and sincos𝜃+𝜃=1, find the value of 𝜃.

Q19:

Find the set of values satisfying 3𝜃2𝜃𝜃=0sinsincos where 0𝜃<360. Give the answer to the nearest minute.

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

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