Worksheet: Simple Trigonometric Equations

In this worksheet, we will practice solving simple trigonometric equations.

Q1:

What is the general solution of sin𝜃=22?

  • A 𝜋 4 + 2 𝑛 𝜋 or 𝜋4+𝜋+2𝑛𝜋 where 𝑛
  • B 𝜋 6 + 2 𝑛 𝜋 or 𝜋6+𝜋+2𝑛𝜋 where 𝑛
  • C 𝜋 4 + 2 𝑛 𝜋 or 𝜋4+𝜋+2𝑛𝜋 where 𝑛
  • D 𝜋 6 + 2 𝑛 𝜋 or 𝜋6+𝜋+2𝑛𝜋 where 𝑛

Q2:

Find the solution set of tantantantan𝑥+7+𝑥7=1, where 0<𝑥<360.

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

Q3:

Find the solution set of sincoscossin𝑥16𝑥16=22, where 0<𝑥<360.

  • A { 6 1 , 1 5 1 }
  • B { 2 9 , 1 5 1 }
  • C { 6 1 , 1 1 9 }
  • D { 2 9 , 1 1 9 }

Q4:

Find the set of values satisfying 4𝜃1=0sin where 90𝜃360.

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

Q5:

Find the general solution to the equation cot𝜋2𝜃=13.

  • A 2 𝜋 3 + 2 𝑛 𝜋 , where 𝑛
  • B 5 𝜋 6 + 2 𝑛 𝜋 , where 𝑛
  • C 2 𝜋 3 + 𝑛 𝜋 , where 𝑛
  • D 5 𝜋 6 + 𝑛 𝜋 , where 𝑛

Q6:

Find the set of values satisfying cos(𝜃105)=12 where 0<𝜃<360.

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

Q7:

Find 𝜃 in degrees given cos(90+𝜃)=12 where 𝜃 is the smallest positive angle.

Q8:

Find the set of values satisfying 2𝜃𝜃𝜃=0sincoscos where 0𝜃<360.

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

Q9:

What is the general solution of cos𝜃=32?

  • A 𝜋 3 + 2 𝑛 𝜋 or 𝜋3+2𝑛𝜋 where 𝑛 is an integer.
  • B 𝜋 4 + 2 𝑛 𝜋 or 𝜋4+2𝑛𝜋 where 𝑛 is an integer.
  • C 𝜋 2 + 2 𝑛 𝜋 or 𝜋2+2𝑛𝜋 where 𝑛 is an integer.
  • D 𝜋 6 + 2 𝑛 𝜋 or 𝜋6+2𝑛𝜋 where 𝑛 is an integer.

Q10:

Find the set of values satisfying 11𝜃+13=0tan where 0𝜃<360. Give the answers to the nearest second.

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

Q11:

Find the values of 𝜃 that satisfy 0<𝜃<360 where tansincos𝜃=1944+6742 giving the answer to the nearest minute.

  • A 3 5 3 9 , 1 4 4 2 1
  • B 3 5 3 9 , 3 2 4 2 1
  • C 1 4 4 2 1 , 2 1 5 3 9
  • D 3 5 3 9 , 2 1 5 3 9

Q12:

Find all the possible general solutions of 2𝜃=3𝜃sinsin.

  • A 𝜋 + 2 𝑛 𝜋 , 2 𝑛 𝜋 , 𝜋 3 + 2 𝑛 𝜋 , 𝜋 3 + 𝜋 + 2 𝑛 𝜋
  • B 𝜋 + 𝑛 𝜋 , 2 𝑛 𝜋 , 𝜋 3 + 2 𝑛 𝜋 , 𝜋 3 + 𝜋 + 2 𝑛 𝜋
  • C 𝜋 + 2 𝑛 𝜋 , 𝑛 𝜋 , 𝜋 3 + 2 𝑛 𝜋 , 𝜋 3 + 𝜋 + 2 𝑛 𝜋
  • D 𝜋 3 + 2 𝑛 𝜋 , 𝜋 3 + 𝜋 + 2 𝑛 𝜋
  • E 𝜋 + 2 𝑛 𝜋 , 2 𝑛 𝜋 , 𝜋 3 + 2 𝑛 𝜋 , 𝜋 3 + 𝜋 + 𝑛 𝜋

Q13:

Find the set of values satisfying sin3𝑥=1, where 0𝑥<2𝜋.

  • A 0 , 2 𝜋 3
  • B 𝜋 6 , 5 𝜋 6
  • C 𝜋 2 , 3 𝜋 2
  • D 𝜋 6 , 5 𝜋 6 , 3 𝜋 2
  • E 𝜋 6 , 2 𝜋

Q14:

Find the general solution to the equation cos(90𝜃)=22.

  • A 𝜋 4 + 2 𝜋 𝑛 or 3𝜋4+2𝜋𝑛 where 𝑛
  • B 𝜋 4 + 2 𝜋 𝑛 or 3𝜋4+2𝜋𝑛 where 𝑛
  • C 𝜋 4 + 2 𝜋 𝑛 or 3𝜋4+2𝜋𝑛 where 𝑛
  • D 𝜋 4 + 2 𝜋 𝑛 or 3𝜋4+2𝜋𝑛 where 𝑛

Q15:

Find the set of values satisfying sin15𝜃7=12 given 0<15𝜃7<360.

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

Q16:

Is there a value of the tangent function that is obtained from ONLY one angle in the interval [0,2𝜋)? If yes, give the angle.

  • Ayes, 𝜋
  • Byes, 𝜋4
  • Cno
  • Dyes, 0
  • Eyes, 𝜋2

Q17:

Find the values of 𝜃 that satisfy 𝜃(0,2𝜋) given csc𝜃=3.3069. Give the answer to the nearest minute.

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

Q18:

Find all the possible values of 𝜃 given sec𝜃=1.245 where 𝜃(0,2𝜋). Give the answer to the nearest second.

  • A 𝜃 = 1 2 6 3 3 4 3 or 𝜃=2332617
  • B 𝜃 = 2 1 6 3 3 4 3 or 𝜃=3232617
  • C 𝜃 = 3 6 3 3 4 3 or 𝜃=3232617
  • D 𝜃 = 3 6 3 3 4 8 or 𝜃=1432617

Q19:

Find all the possible values of 𝜃 given tan𝜃=0.4459 where 𝜃(0,2𝜋). Give the answer to the nearest second.

  • A 𝜃 = 1 1 4 1 5 6 or 𝜃=245584
  • B 𝜃 = 2 0 4 1 5 6 or 𝜃=335584
  • C 𝜃 = 2 4 1 5 6 or 𝜃=204156
  • D 𝜃 = 1 5 5 5 8 4 or 𝜃=204156

Q20:

Find the set of values satisfying tan2𝑥+𝜋5=1, where 0𝑥<2𝜋.

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

Q21:

Find the general solution to the equation sec𝜃=2.

  • A 𝜋 2 + 2 𝜋 𝑛 , 𝜋 2 + 2 𝜋 𝑛 , where 𝑛
  • B 𝜋 4 + 2 𝜋 𝑛 , 𝜋 4 + 2 𝜋 𝑛 , where 𝑛
  • C 3 𝜋 4 + 2 𝜋 𝑛 , 3 𝜋 4 + 2 𝜋 𝑛 , where 𝑛
  • D 2 𝜋 3 + 2 𝜋 𝑛 , 2 𝜋 3 + 2 𝜋 𝑛 , where 𝑛

Q22:

Find the possible values of 𝜃 in the expression 173(360𝛼)+(270𝜃)=3coscot where 0<𝜃<360, given sin𝛼=45 where 180𝛼<270. Give the answer to the nearest second.

  • A 𝜃 = 1 5 8 1 1 5 5 or 𝜃=201485
  • B 𝜃 = 1 5 8 1 1 5 5 or 𝜃=3381155
  • C 𝜃 = 2 1 4 8 5 or 𝜃=3381155
  • D 𝜃 = 2 1 4 8 5 or 𝜃=201485

Q23:

Find the set of values satisfying sin2𝑥+𝜋3=22, where 0𝑥<2𝜋.

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

Q24:

𝐴 𝐵 𝐶 is a triangle where 𝑎=10.1cm, 𝑐=33.1cm and the area is 83.5775 cm. Find all the possible values for 𝑚𝐵 giving the answer to the nearest degree.

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

Q25:

Find 𝑚𝜃 in terms of 𝜋 given 28𝜃=𝜃𝜃+𝜋2costancotcos where 𝜃0,𝜋2.

  • A 𝜋 3 2
  • B 𝜋 6 4
  • C 𝜋 1 2 8
  • D 𝜋 1 6

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