Worksheet: Solving a Trigonometric Equation

In this worksheet, we will practice solving a trigonometric equation using factoring or squaring.

Q1:

Find the set of values satisfying 6𝜃7𝜃5=0coscos where 0𝜃<360. Give the answers to the nearest minute.

  • A{120,300}
  • B{120,240}
  • C{60,300}
  • D{60,240}

Q2:

Find the set of values satisfying 5𝜃=4cos where 0𝜃<360. Give the answer to the nearest minute.

  • A{7326,10634,25326,28634}
  • B{2634,15326,20634,33326}
  • C{3634,14326,21634,32326}
  • D{6326,11634,24326,29634}

Q3:

Find all the possible general solutions of 2𝜃2𝜃=0coscos.

  • A𝜋2+𝑛𝜋,𝜋2+𝑛𝜋,𝜋4+2𝑛𝜋,𝜋4+2𝑛𝜋𝑛:.
  • B𝜋2+2𝑛𝜋,𝜋2+2𝑛𝜋,𝜋4+2𝑛𝜋,𝜋4+2𝑛𝜋𝑛:.
  • C𝜋2+𝑛𝜋,𝜋4+2𝑛𝜋𝑛:.
  • D𝜋2+𝑛𝜋,𝜋4+2𝑛𝜋𝑛:.
  • E𝜋2+𝑛𝜋,𝜋4+2𝑛𝜋𝑛:.

Q4:

Find the set of values satisfying 13𝜃76𝜃=0tantan where 0𝜃<360. Give the answers to the nearest second.

  • A{000,18000,2601736,994224}
  • B{000,18000,801736,2601736}
  • C{000,18000,801736,994224}
  • D{801736,2601736}
  • E{000,18000,36000,801736,2601736}

Q5:

Find the set of values satisfying 2𝜃2𝜃2=0sinsin given 180𝜃<360.

  • A{135,315}
  • B{225,315}
  • C{135,225}
  • D{45,135}

Q6:

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

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

Q7:

Find the set of values satisfying tantan𝜃+𝜃=0 where 0𝜃<180.

  • A{135,45,90,270}
  • B{135,315,0,180}
  • C{45,135,0,90}
  • D{135,225,0,180}

Q8:

Find the set of values satisfying 22𝜃+2𝜃=0coscos given 0<𝜃360.

  • A{45,90,270,315}
  • B{0,45,135,180}
  • C{0,135,180,225}
  • D{90,135,225,270}

Q9:

Find all the possible general solutions of 2𝜃3𝜃=0coscos.

  • A𝜋2+𝑛𝜋,𝜋2+𝑛𝜋,𝜋6+2𝑛𝜋,𝜋6+2𝑛𝜋𝑛:.
  • B𝜋2+2𝑛𝜋,𝜋2+2𝑛𝜋,𝜋6+2𝑛𝜋,𝜋6+2𝑛𝜋𝑛:.
  • C𝜋2+𝑛𝜋,𝜋6+2𝑛𝜋𝑛:.
  • D𝜋2+𝑛𝜋,𝜋6+2𝑛𝜋𝑛:.
  • E𝜋2+𝑛𝜋,𝜋6+2𝑛𝜋𝑛:.

Q10:

Find the set of values satisfying 71𝜃+80𝜃=0tantan where 0𝜃<360. Give the answers to the nearest second.

  • A{482439,1313521}
  • B{000,18000,1313521,3113521}
  • C{000,18000,482439,3113521}
  • D{1313521,3113521}
  • E{482439,2282439}

Q11:

Find the set of values satisfying 78𝜃+49𝜃=0tantan where 0𝜃<360. Give the answers to the nearest second.

  • A{32814,1475146}
  • B{000,18000,1475146,3275146}
  • C{000,18000,32814,3275146}
  • D{1475146,3275146}
  • E{32814,212814}

Q12:

Find the set of values satisfying 6𝜃𝜃1=0coscos where 0𝜃<360. Give the answers to the nearest minute.

  • A{120,300,10928,28928}
  • B{60,10928,300,25032}
  • C{120,240,7032,28928}
  • D{60,300,7032,25032}

Q13:

Consider the equation sincos𝜃+𝜃=2, where 0<𝜃360. Call this Equation A.

Create Equation B by squaring both sides of Equation A. Use the fact that sincos𝜃+𝜃=1 to simplify Equation B.

  • Asincos𝜃𝜃=2
  • Bsincos𝜃𝜃=1
  • Csincos𝜃𝜃=1
  • D2𝜃𝜃=1sincos
  • E2𝜃𝜃=1sincos

Now, use a double angle formula to further simplify Equation B.

  • Asin𝜃=1
  • Bsin2𝜃=2
  • Ccos𝜃=1
  • Dsin2𝜃=1
  • Ecos2𝜃=1

The solutions to Equation A are a subset of the solutions of Equation B. Using this, solve Equation A over the specified range.

  • A𝜃=135
  • B𝜃=60
  • C𝜃=30
  • D𝜃=45
  • E𝜃=215

Q14:

Solve 2𝜃+3𝜃=2sincos, where 0<𝜃2𝜋. Give your answer in radians to two decimal places.

  • A𝜃=0.40,2.95
  • B𝜃=0.22,1.15
  • C𝜃=0.24,1.86
  • D𝜃=0.47,2.17
  • E𝜃=1.69,2.14

Q15:

If 𝜃[0,180) and sincos𝜃+𝜃=1, find the possible values of 𝜃.

  • A0, 90
  • B90, 180
  • C45, 90
  • D0, 180
  • E0, 45

Q16:

By first squaring both sides, or otherwise, solve the equation 4𝜃4𝜃=3sincos, where 0<𝜃360. Be careful to remove any extraneous solutions. Give your answers to two decimal places.

  • A𝜃=86.14,212.57
  • B𝜃=65.18,205.14
  • C𝜃=47.35,195.12
  • D𝜃=77.24,210.57
  • E𝜃=62.83,207.17

Q17:

Find the set of values satisfying 97𝜃+60𝜃=0sincos where 0<𝜃<360. Give the answers to the nearest second.

  • A{314421,3281539}
  • B{1481539,2114421}
  • C{1481539,3281539}
  • D{314421,2114421}
  • E{314421,1481539}

Q18:

If 𝜃(180,360) and sincos𝜃+𝜃=1, find the value of 𝜃.

Q19:

Find the set of values satisfying 3𝜃2𝜃𝜃=0sinsincos where 0𝜃<360. Give the answer to the nearest minute.

  • A{0,3341,180,14619}
  • B{0,14619,180,32619}
  • C{0,3341,180,21341}
  • D{0,14619,180,21341}

Q20:

If tantan3𝑥(90+2𝑥)=1, then the general solution is , where 𝑛.

  • A𝑥=90+360𝑛
  • B𝑥=90+180𝑛
  • C𝑥=18+36𝑛
  • D𝑥=90+180𝑛

Q21:

If 0𝑥360, then the number of solutions of the equation 4𝑥=𝑥sintan is .

  • A2
  • B3
  • C5
  • D4

Q22:

The number of solutions of the equation coscos𝑥4𝑥+4=0 is .

  • A1
  • B3
  • Czero
  • D2

Q23:

The solution set of the equation sincos𝑥+𝑥=0, where 0<𝑥<180, is .

  • A{45}
  • B{135}
  • C{120}
  • D{150}

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