Worksheet: Solving Trigonometric Equations Using Trigonometric Identities

In this worksheet, we will practice using Pythagorean trigonometric identities to solve trigonometric equations.

Q1:

Find the general solution of the equation sincos𝜃=5𝜃.

  • A𝜋12𝜋𝑛3 or 𝜋8+𝜋𝑛2, where 𝑛
  • B𝜋12+𝜋𝑛3 or 𝜋8𝜋𝑛2, where 𝑛
  • C𝜋6+𝜋𝑛6 or 𝜋4𝜋𝑛4, where 𝑛
  • D𝜋3+𝜋𝑛3 or 𝜋2+𝜋𝑛2, where 𝑛

Q2:

Find the value of 𝑋 which gives the maximum value of the expression sincoscossin𝑋61+𝑋61, where 0<𝑋<360.

Q3:

Find the solution set for 𝑥 given sincoscossin𝑥35+𝑥35=22 where 0<𝑥<360.

  • A{10, 170}
  • B{80, 170}
  • C{80, 100}
  • D{10, 100}

Q4:

Find the value of 𝑋 which gives the minimum value of the expression sincoscossin𝑋12+𝑋12, where 0<𝑋<360.

Q5:

Find the measure of 𝜃 given cscsec(𝜃+1524)=(𝜃+3754) where 𝜃 is a positive acute angle. Give the answer to the nearest minute.

  • A1821
  • B3642
  • C4142
  • D2321

Q6:

Find all possible solutions of 𝜃 that satisfy cossin2𝜃6𝜃=0.

  • A𝜋4+𝜋𝑛4,𝜋2+𝜋𝑛2𝑛
  • B𝜋16𝜋𝑛4,𝜋8+𝜋𝑛2𝑛
  • C𝜋16+𝜋𝑛4,𝜋8+𝜋𝑛2𝑛
  • D𝜋8+𝜋𝑛8,𝜋4𝜋𝑛4𝑛

Q7:

Find the value of 𝜃 given cossin32𝜃𝜃=0 where 𝜃 is a positive acute angle.

Q8:

Find the value of 𝜃 given tancot𝜃𝜋5=𝜃 where 𝜃0,𝜋2. Give the answer to the nearest second.

  • A5800
  • B2700
  • C6300
  • D2200

Q9:

Knowing that 5+4𝑥=12𝑥costan, find tan𝑥.

  • Atan𝑥=32
  • Btan𝑥=52
  • Ctan𝑥=12
  • Dtan𝑥=52
  • Etan𝑥=32

Q10:

Find sincos3𝜃+6𝜃 given cottan𝜃=2𝜃 where 𝜃 is a positive acute angle.

  • A12
  • B1
  • C14
  • D0

Q11:

Find the solution set of the equation sinsinsinsin(67+2𝜃)(79+𝜃)+(232𝜃)(11𝜃)=1 given 0<𝜃<𝜋2.

  • A{90}
  • B{34}
  • C{146}
  • D{12}

Q12:

Knowing that 5𝑥+12𝑥=13sincos, find sin𝑥 and cos𝑥.

  • Asincos𝑥=513,𝑥=1213
  • Bsincos𝑥=1312,𝑥=135
  • Csincos𝑥=513,𝑥=1213
  • Dsincos𝑥=1213,𝑥=513
  • Esincos𝑥=135,𝑥=1312

Q13:

Find the value of 𝑋 in degrees given cos(𝑋+19)=12 where 𝑋+19 is an acute angle.

Q14:

Consider the equation sincos𝜃3𝜃=1, where 0<𝜃2𝜋. We can solve this using the addition formula: 𝑅(𝜃𝛼)=𝑅𝜃𝛼𝑅𝜃𝛼sinsincoscossin.

If we compare sincos𝜃3𝜃=1 to the right hand side of the addition formula we find that, for them to be equal, 𝑅𝛼=1cos and that 𝑅𝛼=3sin. Use these two equations to find the values of 𝑅 and 𝛼, giving 𝛼 in radians.

  • A𝑅=1, 𝛼=𝜋4
  • B𝑅=1, 𝛼=𝜋2
  • C𝑅=2, 𝛼=𝜋3
  • D𝑅=3, 𝛼=𝜋6
  • E𝑅=2, 𝛼=𝜋6

Using your values for 𝑅 and 𝛼, solve the equation 𝑅(𝜃𝛼)=1sin to find the solutions for the original equation.

  • A𝜃=𝜋4, 𝜋2
  • B𝜃=𝜋2, 𝜋6
  • C𝜃=𝜋2, 7𝜋6
  • D𝜃=𝜋3, 𝜋6
  • E𝜃=3𝜋2, 5𝜋6

Q15:

Find 𝜃 in degrees given cscsec(𝜃5)=(6𝜃10) where 𝜃 is a positive acute angle.

Q16:

Find the set of values satisfying sectantan𝜃𝜃+3𝜃=0 where 0𝜃<360.

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

Q17:

Determine the value of 𝑋 to the nearest second, given that 4𝑋4530=160sincossincos, where 𝑋 is acute.

  • A275618
  • B32140
  • C851311
  • D575820

Q18:

Find the solution set of 𝑥 given tantantantan(8𝑥40)(5𝑥40)1+(8𝑥40)(5𝑥40)=1 where 0<𝑥<90.

  • A{3,17}
  • B{15,75}
  • C{45,75}
  • D{20,80}

Q19:

Find the value of 𝑋 given cossin(𝑋+6)=51 where (𝑋+6) is an acute angle.

Q20:

Find the value of 𝑋 without using a calculator, given 𝑋𝜋64𝜋3𝜋4=7𝜋63𝜋4sincoscottancos.

  • A23
  • B54
  • C23
  • D54

Q21:

Find the solution set of 𝜃 given tantantantan25𝜃23𝜃1+25𝜃23𝜃=3 where 0<𝜃<90.

  • A{45}
  • B{30}
  • C{60}
  • D{15}

Q22:

Find the value of 𝑋 that gives the minimum value of 𝑦 given 𝑦=𝑋31𝑋31sincoscossin where 0<𝑋<2𝜋.

Q23:

Find the value of 𝑋 that gives the maximum value of 𝑦 given 𝑦=𝑋9𝑋9sincoscossin where 0<𝑋<2𝜋.

Q24:

Find the set of solutions for 𝑥 given sincoscossincos198132+198132=𝑥, where 0<𝑥<360.

  • A{120,240}
  • B{24,336}
  • C{120,210}
  • D{150,210}

Q25:

Find the set of values satisfying 3𝜃+1𝜃=0tancot where 0𝜃<360.

  • A{0,30,180,210}
  • B{0,150,180,330}
  • C{0,30,180,150}
  • D{0,60,180,240}

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