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Worksheet: Solving Trigonometric Equations

Q1:

Find the value of 𝑋 which gives the maximum value of the equation s i n c o s c o s s i n 𝑋 6 1 + 𝑋 6 1 ∘ ∘ where 0 < 𝑋 < 2 πœ‹ .

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

Q2:

Suppose 𝑃 is a point on a unit circle corresponding to the angle of 4 πœ‹ 3 . Is there another point on the unit circle representing an angle in the interval [ 0 , 2 πœ‹ ) that has the same tangent value? If yes, give the angle.

  • Ayes, πœ‹ 6
  • Bno
  • Cyes, πœ‹ 4
  • Dyes, πœ‹ 3
  • Eyes, 1 1 πœ‹ 6

Q3:

Find all the possible general solutions of .

  • A , , ,
  • B , , ,
  • C , , ,
  • D , , ,
  • E ,

Q4:

Find the value of 𝑋 which gives the minimum value of the equation s i n c o s c o s s i n 𝑋 1 2 + 𝑋 1 2 ∘ ∘ where 0 < 𝑋 < 2 πœ‹ .

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

Q5:

Find the general solution to the equation s i n c o s 2 π‘₯ = π‘₯ 2 .

  • A π‘₯ = πœ‹ 2 + 2 πœ‹ 𝑛 , π‘₯ = πœ‹ 5 + 4 𝑛 πœ‹ 5 , where 𝑛 ∈ β„€
  • B π‘₯ = πœ‹ 2 + 2 πœ‹ 𝑛 , π‘₯ = πœ‹ 3 + 4 𝑛 πœ‹ 3 , where 𝑛 ∈ β„€
  • C π‘₯ = πœ‹ 2 + 2 πœ‹ 𝑛 , π‘₯ = πœ‹ + 2 πœ‹ 𝑛 , where 𝑛 ∈ β„€
  • D π‘₯ = πœ‹ 5 + 4 𝑛 πœ‹ 5 , π‘₯ = πœ‹ 3 + 4 𝑛 πœ‹ 3 , where 𝑛 ∈ β„€
  • E π‘₯ = πœ‹ + 2 πœ‹ 𝑛 , π‘₯ = πœ‹ 3 + 4 𝑛 πœ‹ 3 , where 𝑛 ∈ β„€

Q6:

Find the value of 𝐴 given c o s t a n 𝐴 𝐴 = 7 1 2 where 𝐴 is an acute angle. Give the answer to the nearest second.

  • A 3 0 1 5 β€² 2 3 β€² β€² ∘
  • B 5 4 1 8 β€² 5 3 β€² β€² ∘
  • C 5 9 4 4 β€² 3 7 β€² β€² ∘
  • D 3 5 4 1 β€² 7 β€² β€² ∘

Q7:

Consider 𝐴 , a point on a unit circle corresponding to the angle of 3 πœ‹ 2 . Is there another point on the unit circle that has the same 𝑦 -coordinate as 𝐴 and represents an angle in the interval [ 0 , 2 πœ‹ ) ? If yes, give the angle.

  • Ayes, πœ‹ 6
  • Byes, πœ‹ 2
  • Cyes, πœ‹ 3
  • Dno
  • Eyes, πœ‹ 4

Q8:

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

  • Ayes, 0
  • Byes, πœ‹ 4
  • Cyes, πœ‹
  • Dno
  • Eyes, πœ‹ 2

Q9:

Find the possible values of in the expression given where is the greatest angle in the range . Give the angle to the nearest minute.

  • A or
  • B or
  • C or
  • D or

Q10:

Find the value of 𝑋 given c o s 2 𝑋 = √ 3 2 where 2 𝑋 is an acute angle. Give the answer to the nearest minute.

  • A 2 2 3 0 β€² ∘
  • B 3 0 ∘
  • C 4 5 ∘
  • D 1 5 ∘

Q11:

Find the value of 𝑋 given t a n 3 𝑋 = √ 3 where 3 𝑋 is an acute angle. Give the answer to the nearest minute.

  • A 1 5 ∘
  • B 1 0 ∘
  • C 3 0 ∘
  • D 2 0 ∘

Q12:

Find the value of 𝑋 given s i n 3 𝑋 = 1 2 where 3 𝑋 is an acute angle. Give the answer to the nearest minute.

  • A 1 5 ∘
  • B 2 0 ∘
  • C 3 0 ∘
  • D 1 0 ∘

Q13:

Find the possible values of πœƒ in the expression βˆ’ 1 7 3 ( 3 6 0 βˆ’ 𝛼 ) + ( 2 7 0 βˆ’ πœƒ ) = 3 c o s c o t ∘ ∘ where 0 < πœƒ < 3 6 0 ∘ ∘ , given s i n 𝛼 = βˆ’ 4 5 where 1 8 0 ≀ πœƒ < 2 7 0 ∘ ∘ . Give the answer to the nearest second.

  • A πœƒ = 2 1 4 8 β€² 5 β€² β€² ∘ or πœƒ = 2 0 1 4 8 β€² 5 β€² β€² ∘
  • B πœƒ = 2 1 4 8 β€² 5 β€² β€² ∘ or πœƒ = 3 3 8 1 1 β€² 5 5 β€² β€² ∘
  • C πœƒ = 1 5 8 1 1 β€² 5 5 β€² β€² ∘ or πœƒ = 2 0 1 4 8 β€² 5 β€² β€² ∘
  • D πœƒ = 1 5 8 1 1 β€² 5 5 β€² β€² ∘ or πœƒ = 3 3 8 1 1 β€² 5 5 β€² β€² ∘

Q14:

Consider 𝑀 , a point on a unit circle corresponding to the angle of πœ‹ . Is there another point on the unit circle representing an angle in the interval [ 0 , 2 πœ‹ ) that has the same π‘₯ -coordinate as 𝑀 ? If yes, give the angle.

  • Ayes, πœ‹ 2
  • Byes, πœ‹ 3
  • Cyes, πœ‹ 6
  • Dno
  • Eyes, πœ‹ 4

Q15:

Find the solution set of given where .

  • A
  • B
  • C
  • D

Q16:

Find the value of 𝑋 that gives the minimum value of 𝑦 given 𝑦 = 𝑋 3 1 βˆ’ 𝑋 3 1 s i n c o s c o s s i n ∘ ∘ where 0 < 𝑋 < 2 πœ‹ .

Q17:

Find the value of 𝑋 that gives the maximum value of 𝑦 given 𝑦 = 𝑋 9 βˆ’ 𝑋 9 s i n c o s c o s s i n ∘ ∘ where 0 < 𝑋 < 2 πœ‹ .

Q18:

Find the set of values satisfying s i n c o s t a n 1 5 7 5 + √ 3 ( 9 0 + πœƒ ) = 0 ∘ ∘ ∘ , where 0 ≀ πœƒ < 3 6 0 ∘ ∘ .

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

Q19:

Suppose 𝐿 is a point on a unit circle corresponding to the angle of πœ‹ 3 . Is there another point on the unit circle that represents an angle in the interval [ 0 , 2 πœ‹ ) and has the same π‘₯ -coordinate as 𝐿 ? If yes, give the angle.

  • Ano
  • Byes, βˆ’ πœ‹ 6
  • Cyes, 2 πœ‹ 3
  • Dyes, 5 πœ‹ 3
  • Eyes, 7 πœ‹ 1 2

Q20:

Find the set of solutions for π‘₯ given s i n c o s c o s s i n c o s 1 9 8 1 3 2 + 1 9 8 1 3 2 = π‘₯ ∘ ∘ ∘ ∘ , where 0 < π‘₯ < 3 6 0 ∘ ∘ .

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

Q21:

Find the set of values satisfying √ 3 πœƒ + 1 πœƒ = 0 t a n c o t 2 where 0 ≀ πœƒ < 3 6 0 ∘ ∘ .

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

Q22:

Find the set of solutions for π‘₯ given s i n c o s c o s s i n 9 π‘₯ 4 π‘₯ βˆ’ 9 π‘₯ 4 π‘₯ = √ 2 2 where 0 < π‘₯ < 2 πœ‹ 5 ∘ .

  • A { 9 , 3 0 } ∘ ∘
  • B { 6 , 3 0 } ∘ ∘
  • C { 6 , 2 7 } ∘ ∘
  • D { 9 , 2 7 } ∘ ∘

Q23:

Find the set of values satisfying 3 πœƒ βˆ’ 2 πœƒ πœƒ = 0 s i n s i n c o s 2 where 0 ≀ πœƒ < 3 6 0 ∘ ∘ . Give the answer to the nearest minute.

  • A { 0 , 1 4 6 1 9 β€² , 1 8 0 , 2 1 3 4 1 β€² } ∘ ∘ ∘ ∘
  • B { 0 , 3 3 4 1 β€² , 1 8 0 , 1 4 6 1 9 β€² } ∘ ∘ ∘ ∘
  • C { 0 , 1 4 6 1 9 β€² , 1 8 0 , 3 2 6 1 9 β€² } ∘ ∘ ∘ ∘
  • D { 0 , 3 3 4 1 β€² , 1 8 0 , 2 1 3 4 1 β€² } ∘ ∘ ∘ ∘

Q24:

Consider that 𝑁 is a point on a unit circle corresponding to the angle of 5 πœ‹ 6 . Is there another point on the unit circle that has the same 𝑦 -coordinate as 𝑁 and represents an angle in the interval [ 0 , 2 πœ‹ ) ? If yes, give the angle.

  • Ayes, πœ‹ 3
  • Bno
  • Cyes, 7 πœ‹ 6
  • Dyes, πœ‹ 6
  • Eyes, πœ‹ 4

Q25:

Find the value of πœƒ given 2 7 0 < πœƒ < 3 6 0 ∘ ∘ , s i n 𝛼 = βˆ’ 8 1 7 where 2 7 0 < 𝛼 < 3 6 0 ∘ ∘ and t a n 𝛽 = βˆ’ 4 3 where 9 0 < 𝛽 < 1 8 0 ∘ ∘ , and s i n s i n c o s c o s πœƒ = ( 1 8 0 βˆ’ 𝛼 ) ( 𝛽 βˆ’ 1 8 0 ) 𝛼 ∘ ∘ . Give the answer to the nearest minute.

  • A 1 6 5 3 4 β€² ∘
  • B 1 4 2 6 β€² ∘
  • C 1 9 4 2 6 β€² ∘
  • D 3 4 5 3 4 β€² ∘