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.

  • A 𝜋 3 + 𝜋 𝑛 3 or 𝜋2+𝜋𝑛2, where 𝑛
  • B 𝜋 6 + 𝜋 𝑛 6 or 𝜋4𝜋𝑛4, where 𝑛
  • C 𝜋 1 2 + 𝜋 𝑛 3 or 𝜋8𝜋𝑛2, where 𝑛
  • D 𝜋 1 2 𝜋 𝑛 3 or 𝜋8+𝜋𝑛2, where 𝑛

Q2:

Find the value of 𝑋 which gives the maximum value of the equation sincoscossin𝑋61+𝑋61 where 0<𝑋<2𝜋.

  • A 2 0 9
  • B 1 5 1
  • C 2 9
  • D 6 1

Q3:

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

  • A { 1 0 , 1 7 0 }
  • B { 8 0 , 1 0 0 }
  • C { 8 0 , 1 7 0 }
  • D { 1 0 , 1 0 0 }

Q4:

Find the value of 𝑋 which gives the minimum value of the equation sincoscossin𝑋12+𝑋12 where 0<𝑋<2𝜋.

  • A 7 8
  • B 1 0 2
  • C 1 2
  • D 2 5 8

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.

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

Q6:

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

  • A 𝜋 1 6 𝜋 𝑛 4 , 𝜋 8 + 𝜋 𝑛 2 𝑛
  • B 𝜋 4 + 𝜋 𝑛 4 , 𝜋 2 + 𝜋 𝑛 2 𝑛
  • C 𝜋 8 + 𝜋 𝑛 8 , 𝜋 4 𝜋 𝑛 4 𝑛
  • D 𝜋 1 6 + 𝜋 𝑛 4 , 𝜋 8 + 𝜋 𝑛 2 𝑛

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.

  • A 6 3 0 0
  • B 5 8 0 0
  • C 2 7 0 0
  • D 2 2 0 0

Q9:

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

  • A t a n 𝑥 = 1 2
  • B t a n 𝑥 = 5 2
  • C t a n 𝑥 = 3 2
  • D t a n 𝑥 = 3 2
  • E t a n 𝑥 = 5 2

Q10:

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

  • A1
  • B0
  • C 1 2
  • D 1 4

Q11:

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

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

Q12:

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

  • A s i n c o s 𝑥 = 1 2 1 3 , 𝑥 = 5 1 3
  • B s i n c o s 𝑥 = 5 1 3 , 𝑥 = 1 2 1 3
  • C s i n c o s 𝑥 = 5 1 3 , 𝑥 = 1 2 1 3
  • D s i n c o s 𝑥 = 1 3 1 2 , 𝑥 = 1 3 5
  • E s i n c o s 𝑥 = 1 3 5 , 𝑥 = 1 3 1 2

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.

  • A 𝑅 = 3 , 𝛼 = 𝜋 6
  • B 𝑅 = 2 , 𝛼 = 𝜋 3
  • C 𝑅 = 1 , 𝛼 = 𝜋 4
  • D 𝑅 = 2 , 𝛼 = 𝜋 6
  • E 𝑅 = 1 , 𝛼 = 𝜋 2

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

  • A 𝜃 = 3 𝜋 2 , 5 𝜋 6
  • B 𝜃 = 𝜋 2 , 7 𝜋 6
  • C 𝜃 = 𝜋 2 , 𝜋 6
  • D 𝜃 = 𝜋 4 , 𝜋 2
  • E 𝜃 = 𝜋 3 , 𝜋 6

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

Q17:

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

  • A 3 2 1 4 0
  • B 2 7 5 6 1 8
  • C 8 5 1 3 1 1
  • D 5 7 5 8 2 0

Q18:

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

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

Q19:

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

  • A 1 4 7
  • B 1 2 3
  • C 6
  • D 3 3

Q20:

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

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

Q21:

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

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

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

Q25:

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

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

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