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 0𝜃<180, find the solution set of 2𝜃𝜃𝜃=0sincossin.

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

Q2:

Find 𝜃 in degrees given sincos𝜃=4𝜃 where 𝜃 is a positive acute angle.

Q3:

Find the general solution to the equation cossin3𝑥=𝑥4.

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

Q4:

Find the set of possible solutions of 2𝜃𝜃=0sincos given 𝜃[0,360).

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

Q5:

Find the solution set for 𝑥 given coscos2𝑥+133𝑥=19 where 𝑥(0,2𝜋).

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

Q6:

Find all the possible solutions, that is, the general solution, of the equation sincossin𝜃𝜃=22𝜃.

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

Q7:

If 0𝜃<180, find the solution set of 2𝜃𝜃+𝜃=0sincossin.

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

Q8:

Find the solution set for 𝑥 given coscos2𝑥+53𝑥=7 where 𝑥(0,2𝜋).

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

Q9:

Solve tansin𝑥2=𝑥, where 0𝑥<2𝜋.

  • A 𝑥 0 , 𝜋 2
  • B 𝑥 0 , 𝜋 2 , 3 𝜋 2 , 2 𝜋
  • C 𝑥 0 , 𝜋 4 , 𝜋 , 3 𝜋 4
  • D 𝑥 0 , 𝜋 2 , 3 𝜋 2
  • E 𝑥 0 , 𝜋 4

Q10:

By using the half angle formula sincos𝑥2=1𝑥2, or otherwise, solve the equation sincos𝑥2+𝑥=1, where 0𝑥<2𝜋.

  • A 𝑥 = 0 , 1 3 𝜋 , 5 3 𝜋
  • B 𝑥 = 0 , 1 6 𝜋 , 5 6 𝜋
  • C 𝑥 = 0 , 1 3 𝜋 , 5 6 𝜋
  • D 𝑥 = 0 , 1 3 𝜋
  • E 𝑥 = 0 , 1 6 𝜋

Q11:

Find the solution set for 𝑥 given coscossinsin𝑥2𝑥𝑥2𝑥=12 where 0<𝑥<360.

  • A { 2 0 , 1 1 0 }
  • B { 1 0 , 1 1 0 }
  • C { 1 0 , 1 0 0 }
  • D { 2 0 , 1 0 0 }

Q12:

Find all the possible general solutions of cossincos𝜃𝜃=22𝜃.

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

Q13:

Find the set of values satisfying cos2𝑥=32, where 0𝑥<2𝜋.

  • A 5 𝜋 6 , 7 𝜋 6 , 1 1 𝜋 6
  • B { 0 , 𝜋 }
  • C 5 𝜋 1 2 , 7 𝜋 1 2
  • D 5 𝜋 1 2 , 7 𝜋 1 2 , 1 7 𝜋 1 2 , 1 9 𝜋 1 2
  • E 5 𝜋 6 , 7 𝜋 6

Q14:

Find the value of 𝑥 given cossin2𝑥=3𝑥 where 𝑥 is an acute angle. Give the answer to the nearest degree.

Q15:

Find the value of 𝑋 without using a calculator, given 𝑋7𝜋6𝜋3=2𝜋35𝜋6sincostansin.

  • A 1 2
  • B 1 1 2
  • C 1 1 2
  • D12

Q16:

Find the general solution to the equation sincos2𝑥=𝑥2.

  • A 𝑥 = 𝜋 2 + 2 𝜋 𝑛 , 𝑥 = 𝜋 5 + 4 𝑛 𝜋 5 , where 𝑛
  • B 𝑥 = 𝜋 2 + 2 𝜋 𝑛 , 𝑥 = 𝜋 3 + 4 𝑛 𝜋 3 , where 𝑛
  • C 𝑥 = 𝜋 2 + 2 𝜋 𝑛 , 𝑥 = 𝜋 + 2 𝜋 𝑛 , where 𝑛
  • D 𝑥 = 𝜋 + 2 𝜋 𝑛 , 𝑥 = 𝜋 3 + 4 𝑛 𝜋 3 , where 𝑛
  • E 𝑥 = 𝜋 5 + 4 𝑛 𝜋 5 , 𝑥 = 𝜋 3 + 4 𝑛 𝜋 3 , where 𝑛

Q17:

Find the value of 𝑋 given cos2𝑋=32 where 2𝑋 is an acute angle. Give the answer to the nearest minute.

  • A 2 2 3 0
  • B 3 0
  • C 1 5
  • D 4 5

Q18:

Find the value of 𝑋 in degrees given cossinsintansin3𝑋=30604545 where 3𝑋 is an acute angle.

Q19:

Find the value of 𝜃 given sinsincotcossin𝜃=120780+240750 giving the answer to the nearest second.

  • A 𝜃 = 3 4 5 3 1 2 1 or 𝜃=1942839
  • B 𝜃 = 1 4 2 8 3 9 or 𝜃=1653121
  • C 𝜃 = 1 9 4 2 8 3 9 or 𝜃=1653121
  • D 𝜃 = 1 4 2 8 3 9 or 𝜃=1942839

Q20:

Find the set of possible values of 𝑥 which satisfy 1𝑥𝑥=2coscos where 0<𝑥<360.

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

Q21:

Find 𝑚𝜃 given cossinsincos34.534.5+1269=𝜃 where 𝜃 is a positive acute angle.

Q22:

Find the set of solutions for 𝑥 given sincoscossin9𝑥4𝑥9𝑥4𝑥=22 where 0<𝑥<2𝜋5 .

  • A { 9 , 3 0 }
  • B { 9 , 2 7 }
  • C { 6 , 2 7 }
  • D { 6 , 3 0 }

Q23:

Find the set of solutions in the range 0<𝑥<180 for the equation (𝑥+𝑥)=22𝑥sincossin.

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

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