Worksheet: Solving a Trigonometric Equation by Squaring

In this worksheet, we will practice solving trigonometric equations by squaring both sides.

Q1:

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

Q2:

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

Q3:

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 }

Q4:

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

Q5:

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

Q6:

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

Q7:

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

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

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