Lesson Worksheet: Simple Trigonometric Equations Mathematics

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

Q1:

What is the general solution of sin𝜃=22?

  • A𝜋4+2𝑛𝜋 or 𝜋4+𝜋+2𝑛𝜋 where 𝑛
  • B𝜋6+2𝑛𝜋 or 𝜋6+𝜋+2𝑛𝜋 where 𝑛
  • C𝜋4+2𝑛𝜋 or 𝜋4+𝜋+2𝑛𝜋 where 𝑛
  • D𝜋6+2𝑛𝜋 or 𝜋6+𝜋+2𝑛𝜋 where 𝑛

Q2:

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

  • A{38,218}
  • B{38,232}
  • C{52,232}
  • D{52,218}

Q3:

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

  • A{30}
  • B{30,150}
  • C{30,150,210,330}
  • D{150,210,330}

Q4:

Find the general solution to the equation cot𝜋2𝜃=13.

  • A2𝜋3+2𝑛𝜋, where 𝑛
  • B5𝜋6+2𝑛𝜋, where 𝑛
  • C2𝜋3+𝑛𝜋, where 𝑛
  • D5𝜋6+𝑛𝜋, where 𝑛

Q5:

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

  • A{75,225}
  • B{345,225}
  • C{135,225}
  • D{105,345}
  • E{255,345}

Q6:

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

Q7:

What is the general solution of cos𝜃=32?

  • A𝜋3+2𝑛𝜋 or 𝜋3+2𝑛𝜋 where 𝑛 is an integer.
  • B𝜋4+2𝑛𝜋 or 𝜋4+2𝑛𝜋 where 𝑛 is an integer.
  • C𝜋2+2𝑛𝜋 or 𝜋2+2𝑛𝜋 where 𝑛 is an integer.
  • D𝜋6+2𝑛𝜋 or 𝜋6+2𝑛𝜋 where 𝑛 is an integer.

Q8:

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

  • A{1301411,3101411}
  • B{494549,2294549}
  • C{494549,3101411}
  • D{494549,1301411}
  • E{1301411,2294549}

Q9:

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

  • A3539, 14421
  • B3539, 32421
  • C14421, 21539
  • D3539, 21539

Q10:

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

  • A0,2𝜋3
  • B𝜋6,5𝜋6
  • C𝜋2,3𝜋2
  • D𝜋6,5𝜋6,3𝜋2
  • E𝜋6,2𝜋

Q11:

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

  • A𝜋4+2𝜋𝑛 or 3𝜋4+2𝜋𝑛 where 𝑛
  • B𝜋4+2𝜋𝑛 or 3𝜋4+2𝜋𝑛 where 𝑛
  • C𝜋4+2𝜋𝑛 or 3𝜋4+2𝜋𝑛 where 𝑛
  • D𝜋4+2𝜋𝑛 or 3𝜋4+2𝜋𝑛 where 𝑛

Q12:

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

  • A{63}
  • B{201,339}
  • C{21}
  • D{45,135}
  • E{21,63}

Q13:

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

Q14:

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

  • A{10736,25224}
  • B{19736,34224}
  • C{1736,34224}
  • D{1736,16224}

Q15:

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

  • A𝜃=1263343 or 𝜃=2332617
  • B𝜃=2163343 or 𝜃=3232617
  • C𝜃=363343 or 𝜃=3232617
  • D𝜃=363348 or 𝜃=1432617

Q16:

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

  • A𝜃=114156 or 𝜃=245584
  • B𝜃=204156 or 𝜃=335584
  • C𝜃=24156 or 𝜃=204156
  • D𝜃=155584 or 𝜃=204156

Q17:

Find the set of values satisfying tan2𝑥+𝜋5=1, where 0𝑥2𝜋.

  • A11𝜋40,31𝜋40
  • B11𝜋40,31𝜋40,51𝜋40
  • C{0,2𝜋}
  • D3𝜋4,7𝜋4
  • E11𝜋40,31𝜋40,51𝜋40,71𝜋40

Q18:

Find the general solution to the equation sec𝜃=2.

  • A𝜋2+2𝜋𝑛, 𝜋2+2𝜋𝑛, where 𝑛
  • B𝜋4+2𝜋𝑛, 𝜋4+2𝜋𝑛, where 𝑛
  • C3𝜋4+2𝜋𝑛, 3𝜋4+2𝜋𝑛, where 𝑛
  • D2𝜋3+2𝜋𝑛, 2𝜋3+2𝜋𝑛, where 𝑛

Q19:

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.

  • A𝜃=1581155 or 𝜃=201485
  • B𝜃=1581155 or 𝜃=3381155
  • C𝜃=21485 or 𝜃=3381155
  • D𝜃=21485 or 𝜃=201485

Q20:

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

  • A{0,2𝜋}
  • B5𝜋24,23𝜋24
  • C5𝜋24,23𝜋24,29𝜋24,47𝜋24
  • D𝜋4,5𝜋4
  • E5𝜋24,23𝜋24,29𝜋24

Q21:

𝐴𝐵𝐶 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.

  • A30, 150
  • B30
  • C14, 166
  • D14

Q22:

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

  • A𝜋32
  • B𝜋64
  • C𝜋128
  • D𝜋16

Q23:

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

  • A{30,150,210,330}
  • B{45,135,225,315}
  • C{60,120,240,300}

Q24:

Find the general solution to the equation sincos𝑥𝜋4=2𝑥𝜋3.

  • A𝑥=7𝜋36+2𝑛𝜋3, 𝑥=13𝜋36+2𝑛𝜋3, where 𝑛
  • B𝑥=7𝜋36+2𝑛𝜋3, 𝑥=19𝜋12+2𝑛𝜋, where 𝑛
  • C𝑥=7𝜋36+2𝑛𝜋3, 𝑥=𝜋12+2𝑛𝜋, where 𝑛
  • D𝑥=13𝜋36+𝑛𝜋3, 𝑥=19𝜋12+𝑛𝜋, where 𝑛
  • E𝑥=13𝜋36+2𝑛𝜋3, 𝑥=19𝜋12+2𝑛𝜋, where 𝑛

Q25:

Find the set of values satisfying cos3𝑥+𝜋2=12, where 0𝑥<2𝜋.

  • A{0,2𝜋}
  • B7𝜋18,11𝜋18
  • C𝜋6,10𝜋9
  • D7𝜋18,11𝜋18,19𝜋18,23𝜋18
  • E7𝜋18,11𝜋18,19𝜋18,23𝜋18,31𝜋18,35𝜋18

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