Worksheet: Solving Trigonometric Equations with the Double-Angle Identity

In this worksheet, we will practice solving trigonometric equations using the double angle identity.

Q1:

If 0 𝜃 < 1 8 0 , find the solution set of 2 𝜃 𝜃 𝜃 = 0 s i n c o s s i n .

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

Q2:

Find 𝜃 in degrees given s i n c o s 𝜃 = 4 𝜃 where 𝜃 is a positive acute angle.

Q3:

Find the general solution to the equation c o s s i n 3 𝑥 = 𝑥 4 .

  • A 𝑥 = 2 𝜋 1 3 + 4 𝑛 𝜋 1 3 , 𝑥 = 2 𝜋 1 1 + 4 𝑛 𝜋 1 1 , where 𝑛
  • B 𝑥 = 2 𝜋 1 3 + 2 𝑛 𝜋 1 3 , 𝑥 = 2 𝜋 1 1 + 2 𝑛 𝜋 1 1 , where 𝑛
  • C 𝑥 = 𝜋 1 2 + 𝑛 𝜋 3 , 𝑥 = 2 𝜋 1 1 + 8 𝑛 𝜋 1 1 , where 𝑛
  • D 𝑥 = 2 𝜋 1 3 + 8 𝑛 𝜋 1 3 , 𝑥 = 2 𝜋 1 1 + 8 𝑛 𝜋 1 1 , where 𝑛
  • E 𝑥 = 𝜋 1 2 + 𝑛 𝜋 3 , 𝑥 = 2 𝜋 1 3 + 8 𝑛 𝜋 1 3 , where 𝑛

Q4:

Find the set of possible solutions of 2 𝜃 𝜃 = 0 s i n c o s given 𝜃 [ 0 , 3 6 0 [ .

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

Q5:

Find the solution set for 𝑥 given c o s c o s 2 𝑥 + 1 3 3 𝑥 = 1 9 where 𝑥 ] 0 , 2 𝜋 [ .

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

Q6:

Find all the possible solutions, that is, the general solution, of the equation s i n c o s s i n 𝜃 𝜃 = 2 2 𝜃 .

  • A 𝑛 𝜋 , 𝜋 2 + 2 𝑛 𝜋 (where 𝑛 )
  • B 𝑛 𝜋 , 𝜋 4 + 2 𝑛 𝜋 (where 𝑛 )
  • C 𝑛 𝜋 , ± 𝜋 2 + 2 𝑛 𝜋 (where 𝑛 )
  • D 𝑛 𝜋 , ± 𝜋 4 + 2 𝑛 𝜋 (where 𝑛 )
  • E ± 𝜋 4 + 2 𝑛 𝜋 (where 𝑛 )

Q7:

If 0 𝜃 < 1 8 0 , find the solution set of 2 𝜃 𝜃 + 𝜃 = 0 s i n c o s s i n .

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

Q8:

Find the solution set for 𝑥 given c o s c o s 2 𝑥 + 5 3 𝑥 = 7 where 𝑥 ] 0 , 2 𝜋 [ .

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

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