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Lesson: Trigonometric Equations

In this lesson, we will learn how to solve simple trigonometric equations.

Sample Question Videos

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05:50
04:03

Worksheet • 25 Questions • 5 Videos

Q1:

What is the general solution of s i n πœƒ = √ 2 2 ?

  • A πœ‹ 4 + 2 𝑛 πœ‹ or βˆ’ πœ‹ 4 + πœ‹ + 2 𝑛 πœ‹ where 𝑛 ∈ β„€
  • B πœ‹ 4 + 2 𝑛 πœ‹ or πœ‹ 4 + πœ‹ + 2 𝑛 πœ‹ where 𝑛 ∈ β„€
  • C πœ‹ 6 + 2 𝑛 πœ‹ or πœ‹ 6 + πœ‹ + 2 𝑛 πœ‹ where 𝑛 ∈ β„€
  • D πœ‹ 6 + 2 𝑛 πœ‹ or βˆ’ πœ‹ 6 + πœ‹ + 2 𝑛 πœ‹ where 𝑛 ∈ β„€

Q2:

Find the value of 𝑋 given t a n ο€Ό 𝑋 4  = √ 3 where 𝑋 4 is an acute angle.

Q3:

Find the solution set of t a n t a n t a n t a n π‘₯ + 7 + π‘₯ 7 = 1 ∘ ∘ , where 0 < π‘₯ < 3 6 0 ∘ ∘ .

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

Q4:

Find the solution set of s i n c o s c o s s i n π‘₯ 1 6 βˆ’ π‘₯ 1 6 = √ 2 2 ∘ ∘ , where 0 < π‘₯ < 3 6 0 ∘ ∘ .

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

Q5:

Find the solution set of π‘₯ given t a n t a n t a n t a n π‘₯ βˆ’ 6 4 1 + π‘₯ 6 4 = 1 ∘ ∘ where 0 < π‘₯ < 3 6 0 ∘ ∘ .

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

Q6:

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‹ 3
  • Byes, 1 1 πœ‹ 6
  • Cno
  • Dyes, πœ‹ 4
  • Eyes, πœ‹ 6

Q7:

Consider , a point on a unit circle corresponding to the angle of . Is there another point on the unit circle that has the same -coordinate as and represents an angle in the interval ? If yes, give the angle.

  • Ano
  • Byes,
  • Cyes,
  • Dyes,
  • Eyes,

Q8:

Find the set of values satisfying 4 πœƒ βˆ’ 1 = 0 s i n 2 where 9 0 ≀ πœƒ ≀ 3 6 0 ∘ ∘ .

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

Q9:

Find the general solution to the equation c o t ο€» πœ‹ 2 βˆ’ πœƒ  = βˆ’ 1 √ 3 .

  • A 5 πœ‹ 6 + 𝑛 πœ‹ , where 𝑛 ∈ β„€
  • B 5 πœ‹ 6 + 2 𝑛 πœ‹ , where 𝑛 ∈ β„€
  • C 2 πœ‹ 3 + 2 𝑛 πœ‹ , where 𝑛 ∈ β„€
  • D 2 πœ‹ 3 + 𝑛 πœ‹ , where 𝑛 ∈ β„€

Q10:

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

  • Ayes,
  • Byes,
  • Cyes,
  • Dyes,
  • Eno

Q11:

Find the set of values satisfying c o s ( πœƒ βˆ’ 1 0 5 ) = βˆ’ 1 2 where 0 < πœƒ < 3 6 0 ∘ ∘ .

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

Q12:

Find πœƒ in degrees given c o s ( 9 0 + πœƒ ) = βˆ’ 1 2 ∘ where πœƒ is the smallest positive angle.

Q13:

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

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

Q14:

Find the solution set for π‘₯ given c o s c o s s i n s i n π‘₯ 2 π‘₯ βˆ’ π‘₯ 2 π‘₯ = 1 2 where 0 < π‘₯ < 3 6 0 ∘ ∘ .

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

Q15:

Find the solution set for π‘₯ given s i n c o s c o s s i n π‘₯ 3 5 + π‘₯ 3 5 = √ 2 2 ∘ ∘ where 0 < π‘₯ < 3 6 0 ∘ ∘ .

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

Q16:

Find the solution set of πœƒ given t a n t a n t a n t a n 2 5 πœƒ βˆ’ 2 3 πœƒ 1 + 2 5 πœƒ 2 3 πœƒ = √ 3 where 0 < πœƒ < 9 0 ∘ ∘ .

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

Q17:

Find the solution set of the equation s i n s i n s i n s i n ( 6 7 + 2 πœƒ ) ( 7 9 + πœƒ ) + ( 2 3 βˆ’ 2 πœƒ ) ( 1 1 βˆ’ πœƒ ) = 1 ∘ ∘ ∘ ∘ given 0 < πœƒ < πœ‹ 2 .

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

Q18:

Find the set of possible values of π‘₯ which satisfy 1 √ π‘₯ βˆ’ π‘₯ = 2 c o s c o s 2 4 where 0 < π‘₯ < 3 6 0 ∘ ∘ .

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

Q19:

Find π‘š ∠ πœƒ given c o s s i n s i n c o s 3 4 . 5 3 4 . 5 + 1 2 6 9 = πœƒ ∘ ∘ ∘ where πœƒ is a positive acute angle.

Q20:

Find the set of solutions in the range 0 < π‘₯ < 1 8 0 for the equation ( π‘₯ + π‘₯ ) = 2 2 π‘₯ s i n c o s s i n 2 2 .

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

Q21:

Find the solution set of given , where .

  • A
  • B
  • C
  • D

Q22:

What is the general solution of c o s πœƒ = √ 3 2 ?

  • A πœ‹ 6 + 2 𝑛 πœ‹ or βˆ’ πœ‹ 6 + 2 𝑛 πœ‹ where 𝑛 is an integer.
  • B πœ‹ 2 + 2 𝑛 πœ‹ or βˆ’ πœ‹ 2 + 2 𝑛 πœ‹ where 𝑛 is an integer.
  • C πœ‹ 3 + 2 𝑛 πœ‹ or βˆ’ πœ‹ 3 + 2 𝑛 πœ‹ where 𝑛 is an integer.
  • D πœ‹ 4 + 2 𝑛 πœ‹ or βˆ’ πœ‹ 4 + 2 𝑛 πœ‹ where 𝑛 is an integer.

Q23:

Find the set of values satisfying 1 1 πœƒ + 1 3 = 0 t a n where 0 ≀ πœƒ < 3 6 0 ∘ ∘ . Give the answers to the nearest second.

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

Q24:

Find all the possible general solutions of c o s s i n c o s πœƒ πœƒ = √ 2 2 πœƒ .

  • A 2 𝑛 πœ‹ Β± πœ‹ 2 , πœ‹ 4 + 2 𝑛 πœ‹ , βˆ’ πœ‹ 4 + πœ‹ + 2 𝑛 πœ‹
  • B 2 𝑛 πœ‹ βˆ’ πœ‹ 2 , πœ‹ 4 + 2 𝑛 πœ‹ , βˆ’ πœ‹ 4 + πœ‹ + 2 𝑛 πœ‹
  • C 2 𝑛 πœ‹ Β± πœ‹ 2 , πœ‹ 4 + 2 𝑛 πœ‹ , πœ‹ 4 + πœ‹ + 2 𝑛 πœ‹
  • D 2 𝑛 πœ‹ + πœ‹ 2 , πœ‹ 4 + 2 𝑛 πœ‹ , βˆ’ πœ‹ 4 + πœ‹ + 2 𝑛 πœ‹
  • E 2 𝑛 πœ‹ Β± πœ‹ 2 , πœ‹ 4 + 2 𝑛 πœ‹ , βˆ’ πœ‹ 4 + πœ‹

Q25:

Find the set of values satisfying c o s 2 π‘₯ = βˆ’ √ 3 2 , where 0 ≀ π‘₯ < 2 πœ‹ .

  • A  5 πœ‹ 1 2 , 7 πœ‹ 1 2 , 1 7 πœ‹ 1 2 , 1 9 πœ‹ 1 2 
  • B { 0 , πœ‹ }
  • C  5 πœ‹ 1 2 , 7 πœ‹ 1 2 
  • D  5 πœ‹ 6 , 7 πœ‹ 6 , 1 1 πœ‹ 6 
  • E  5 πœ‹ 6 , 7 πœ‹ 6 
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