Worksheet: Trigonometric Equations

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

Q1:

What is the general solution of s i n 𝜃 = 2 2 ?

  • A 𝜋 6 + 2 𝑛 𝜋 or 𝜋 6 + 𝜋 + 2 𝑛 𝜋 where 𝑛
  • B 𝜋 4 + 2 𝑛 𝜋 or 𝜋 4 + 𝜋 + 2 𝑛 𝜋 where 𝑛
  • C 𝜋 6 + 2 𝑛 𝜋 or 𝜋 6 + 𝜋 + 2 𝑛 𝜋 where 𝑛
  • D 𝜋 4 + 2 𝑛 𝜋 or 𝜋 4 + 𝜋 + 2 𝑛 𝜋 where 𝑛

Q2:

Find the solution set of t a n t a n t a n t a n 𝑥 + 7 + 𝑥 7 = 1 , where 0 < 𝑥 < 3 6 0 .

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

Q3:

Find the solution set of s i n c o s c o s s i n 𝑥 1 6 𝑥 1 6 = 2 2 , where 0 < 𝑥 < 3 6 0 .

  • A { 2 9 , 1 5 1 }
  • B { 2 9 , 1 1 9 }
  • C { 6 1 , 1 1 9 }
  • D { 6 1 , 1 5 1 }

Q4:

Find the solution set of 𝑥 given t a n t a n t a n t a n 𝑥 6 4 1 + 𝑥 6 4 = 1 where 0 < 𝑥 < 3 6 0 .

  • A { 1 0 9 , 1 6 1 }
  • B { 1 9 , 2 8 9 }
  • C { 1 9 , 1 6 1 }
  • D { 1 0 9 , 2 8 9 }

Q5:

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

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

Q6:

Consider 𝐴 , a point on a unit circle corresponding to the angle of 3 𝜋 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 [ ? If yes, give the angle.

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

Q7:

Find the set of values satisfying 4 𝜃 1 = 0 s i n 2 where 9 0 𝜃 3 6 0 .

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

Q8:

Find the general solution to the equation c o t 𝜋 2 𝜃 = 1 3 .

  • A 2 𝜋 3 + 𝑛 𝜋 , where 𝑛
  • B 5 𝜋 6 + 2 𝑛 𝜋 , where 𝑛
  • C 2 𝜋 3 + 2 𝑛 𝜋 , where 𝑛
  • D 5 𝜋 6 + 𝑛 𝜋 , where 𝑛

Q9:

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

  • Ano
  • Byes, 𝜋 6
  • Cyes, 2 𝜋 3
  • Dyes, 5 𝜋 3
  • Eyes, 7 𝜋 1 2

Q10:

Find the set of values satisfying c o s ( 𝜃 1 0 5 ) = 1 2 where 0 < 𝜃 < 3 6 0 .

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

Q11:

Find 𝜃 in degrees given c o s ( 9 0 + 𝜃 ) = 1 2 where 𝜃 is the smallest positive angle.

Q12:

Find the set of values satisfying 2 𝜃 𝜃 𝜃 = 0 s i n c o s c o s where 0 𝜃 < 3 6 0 .

  • A { 4 5 , 1 3 5 , 1 8 0 }
  • B { 4 5 , 9 0 , 3 1 5 }
  • C { 1 8 0 , 2 2 5 , 3 1 5 }
  • D { 4 5 , 9 0 , 1 3 5 }

Q13:

What is the general solution of c o s 𝜃 = 3 2 ?

  • A 𝜋 4 + 2 𝑛 𝜋 or 𝜋 4 + 2 𝑛 𝜋 where 𝑛 is an integer.
  • B 𝜋 2 + 2 𝑛 𝜋 or 𝜋 2 + 2 𝑛 𝜋 where 𝑛 is an integer.
  • C 𝜋 3 + 2 𝑛 𝜋 or 𝜋 3 + 2 𝑛 𝜋 where 𝑛 is an integer.
  • D 𝜋 6 + 2 𝑛 𝜋 or 𝜋 6 + 2 𝑛 𝜋 where 𝑛 is an integer.

Q14:

Find the set of values satisfying 1 1 𝜃 + 1 3 = 0 t a n where 0 𝜃 < 3 6 0 . Give the answers to the nearest second.

  • A { 1 3 0 1 4 1 1 , 2 2 9 4 5 4 9 }
  • B { 4 9 4 5 4 9 , 3 1 0 1 4 1 1 }
  • C { 4 9 4 5 4 9 , 2 2 9 4 5 4 9 }
  • D { 1 3 0 1 4 1 1 , 3 1 0 1 4 1 1 }
  • E { 4 9 4 5 4 9 , 1 3 0 1 4 1 1 }

Q15:

Find the values of 𝜃 that satisfy 0 < 𝜃 < 3 6 0 where t a n s i n c o s 𝜃 = 1 9 4 4 + 6 7 4 2 giving the answer to the nearest minute.

  • A 3 5 3 9 , 3 2 4 2 1
  • B 3 5 3 9 , 1 4 4 2 1
  • C 1 4 4 2 1 , 2 1 5 3 9
  • D 3 5 3 9 , 2 1 5 3 9

Q16:

Find all the possible general solutions of 2 𝜃 = 3 𝜃 s i n s i n .

  • A 𝜋 + 𝑛 𝜋 , 2 𝑛 𝜋 , 𝜋 3 + 2 𝑛 𝜋 , 𝜋 3 + 𝜋 + 2 𝑛 𝜋
  • B 𝜋 + 2 𝑛 𝜋 , 𝑛 𝜋 , 𝜋 3 + 2 𝑛 𝜋 , 𝜋 3 + 𝜋 + 2 𝑛 𝜋
  • C 𝜋 + 2 𝑛 𝜋 , 2 𝑛 𝜋 , 𝜋 3 + 2 𝑛 𝜋 , 𝜋 3 + 𝜋 + 𝑛 𝜋
  • D 𝜋 + 2 𝑛 𝜋 , 2 𝑛 𝜋 , 𝜋 3 + 2 𝑛 𝜋 , 𝜋 3 + 𝜋 + 2 𝑛 𝜋
  • E 𝜋 3 + 2 𝑛 𝜋 , 𝜋 3 + 𝜋 + 2 𝑛 𝜋

Q17:

Find the set of values satisfying s i n 3 𝑥 = 1 , where 0 𝑥 < 2 𝜋 .

  • A 𝜋 2 , 3 𝜋 2
  • B 𝜋 6 , 5 𝜋 6
  • C 0 , 2 𝜋 3
  • D 𝜋 6 , 5 𝜋 6 , 3 𝜋 2
  • E 𝜋 6 , 2 𝜋

Q18:

Find the general solution to the equation c o s ( 9 0 𝜃 ) = 2 2 .

  • A 𝜋 4 + 2 𝜋 𝑛 or 3 𝜋 4 + 2 𝜋 𝑛 where 𝑛
  • B 𝜋 4 + 2 𝜋 𝑛 or 3 𝜋 4 + 2 𝜋 𝑛 where 𝑛
  • C 𝜋 4 + 2 𝜋 𝑛 or 3 𝜋 4 + 2 𝜋 𝑛 where 𝑛
  • D 𝜋 4 + 2 𝜋 𝑛 or 3 𝜋 4 + 2 𝜋 𝑛 where 𝑛

Q19:

Find the set of values satisfying s i n 1 5 𝜃 7 = 1 2 given 0 < 1 5 𝜃 7 < 3 6 0 .

  • A { 6 3 }
  • B { 2 1 }
  • C { 4 5 , 1 3 5 }
  • D { 2 1 , 6 3 }
  • E { 2 0 1 , 3 3 9 }

Q20:

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

  • Ayes, 0
  • Byes, 𝜋 4
  • Cyes, 𝜋
  • Dno
  • Eyes, 𝜋 2

Q21:

Find the values of 𝜃 that satisfy 𝜃 ( 0 , 2 𝜋 ) given c s c 𝜃 = 3 . 3 0 6 9 . Give the answer to the nearest minute.

  • A { 1 0 7 3 6 , 2 5 2 2 4 }
  • B { 1 7 3 6 , 1 6 2 2 4 }
  • C { 1 7 3 6 , 3 4 2 2 4 }
  • D { 1 9 7 3 6 , 3 4 2 2 4 }

Q22:

Find all the possible values of 𝜃 given s e c 𝜃 = 1 . 2 4 5 where 𝜃 ] 0 , 2 𝜋 [ . Give the answer to the nearest second.

  • A 𝜃 = 1 2 6 3 3 4 3 or 𝜃 = 2 3 3 2 6 1 7
  • B 𝜃 = 2 1 6 3 3 4 3 or 𝜃 = 3 2 3 2 6 1 7
  • C 𝜃 = 3 6 3 3 4 8 or 𝜃 = 1 4 3 2 6 1 7
  • D 𝜃 = 3 6 3 3 4 3 or 𝜃 = 3 2 3 2 6 1 7

Q23:

Find all the possible values of 𝜃 given t a n 𝜃 = 0 . 4 4 5 9 where 𝜃 ( 0 , 2 𝜋 ) . Give the answer to the nearest second.

  • A 𝜃 = 1 1 4 1 5 6 or 𝜃 = 2 4 5 5 8 4
  • B 𝜃 = 2 0 4 1 5 6 or 𝜃 = 3 3 5 5 8 4
  • C 𝜃 = 1 5 5 5 8 4 or 𝜃 = 2 0 4 1 5 6
  • D 𝜃 = 2 4 1 5 6 or 𝜃 = 2 0 4 1 5 6

Q24:

Find the set of values satisfying t a n 2 𝑥 + 𝜋 5 = 1 , where 0 𝑥 < 2 𝜋 .

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

Q25:

Find the general solution to the equation s e c 𝜃 = 2 .

  • A 𝜋 4 + 2 𝜋 𝑛 , 𝜋 4 + 2 𝜋 𝑛 , where 𝑛
  • B 𝜋 2 + 2 𝜋 𝑛 , 𝜋 2 + 2 𝜋 𝑛 , where 𝑛
  • C 2 𝜋 3 + 2 𝜋 𝑛 , 2 𝜋 3 + 2 𝜋 𝑛 , where 𝑛
  • D 3 𝜋 4 + 2 𝜋 𝑛 , 3 𝜋 4 + 2 𝜋 𝑛 , where 𝑛

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.