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Lesson: Cofunction Identities

In this lesson, we will learn how to solve trigonometric equations using cofunction identities.

Sample Question Videos

04:55

Worksheet • 25 Questions • 1 Video

Q1:

Find the value of s e c ( 9 0 + πœƒ ) ∘ given c s c πœƒ = 1 7 8 where 0 < πœƒ < 9 0 ∘ ∘ .

  • A βˆ’ 1 7 8
  • B 8 1 7
  • C βˆ’ 8 1 7
  • D 1 7 8

Q2:

Find all the possible general solutions for the equation s i n c o s πœƒ = 5 πœƒ .

  • A πœ‹ 1 2 + πœ‹ 𝑛 3 or βˆ’ πœ‹ 8 βˆ’ πœ‹ 𝑛 2 , where 𝑛 ∈ β„€
  • B πœ‹ 3 + πœ‹ 𝑛 3 or πœ‹ 2 + πœ‹ 𝑛 2 , where 𝑛 ∈ β„€
  • C πœ‹ 1 2 βˆ’ πœ‹ 𝑛 3 or πœ‹ 8 + πœ‹ 𝑛 2 , where 𝑛 ∈ β„€
  • D πœ‹ 6 + πœ‹ 𝑛 6 or πœ‹ 4 βˆ’ πœ‹ 𝑛 4 , where 𝑛 ∈ β„€

Q3:

Find the value of c s c s e c c s c ( 5 6 ) ( 3 4 ) + ( 1 8 0 βˆ’ πœƒ ) ∘ ∘ ∘ given t a n c o t ( πœƒ + 1 0 ) ( πœƒ + 2 0 ) = 1 ∘ ∘ .

  • A2
  • B 1 2
  • C βˆ’ 1 2
  • D βˆ’ 2

Q4:

Find the general solution of the equation s i n ( 9 0 βˆ’ πœƒ ) = 1 2 ∘ .

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

Q5:

Find s i n 6 πœƒ given c o s s i n 2 πœƒ = 7 πœƒ where πœƒ is a positive acute angle.

  • A √ 3 2
  • B 1 2
  • C1
  • D √ 2 2

Q6:

Find the value of c o t ( πœƒ βˆ’ 9 0 ) ∘ given s e c πœƒ = βˆ’ 1 7 1 5 where 9 0 < πœƒ < 1 8 0 ∘ ∘ .

  • A 8 1 5
  • B βˆ’ 8 1 5
  • C βˆ’ 1 5 8
  • D 1 5 8

Q7:

Find the measure of ∠ πœƒ given c s c s e c ( πœƒ + 1 5 2 4 β€² ) = ( πœƒ + 3 7 5 4 β€² ) ∘ ∘ where πœƒ is a positive acute angle. Give the answer to the nearest minute.

  • A 1 8 2 1 β€² ∘
  • B 3 6 4 2 β€² ∘
  • C 4 1 4 2 β€² ∘
  • D 2 3 2 1 β€² ∘

Q8:

Find the value of c o s ( 1 8 0 + πœƒ ) ∘ given s i n ( 9 0 βˆ’ πœƒ ) = βˆ’ 7 1 7 ∘ where πœƒ is the smallest positive angle.

  • A 7 1 7
  • B βˆ’ 7 1 7
  • C βˆ’ 4 √ 1 5 1 7
  • D 4 √ 1 5 1 7

Q9:

Find s i n 8 πœƒ given s e c ( 9 0 + πœƒ ) βˆ’ 2 = 0 ∘ where 1 8 0 < πœƒ < 2 7 0 ∘ ∘ .

  • A √ 3 2
  • B βˆ’ 1 2
  • C βˆ’ √ 3
  • D 1 2

Q10:

Find the value of c o t ο€» πœ‹ 2 βˆ’ 2 𝐡  given t a n 𝐡 = βˆ’ 3 2 where 3 πœ‹ 2 < 𝐡 < 2 πœ‹ .

  • A 1 2 5
  • B 6 5
  • C βˆ’ 5 6
  • D 5 6

Q11:

Find all possible solutions of πœƒ that satisfy c o s s i n 2 πœƒ βˆ’ 6 πœƒ = 0 .

  • A  πœ‹ 1 6 + πœ‹ 𝑛 4 , πœ‹ 8 + πœ‹ 𝑛 2 ∢ 𝑛 ∈ β„€ 
  • B  πœ‹ 4 + πœ‹ 𝑛 4 , πœ‹ 2 + πœ‹ 𝑛 2 ∢ 𝑛 ∈ β„€ 
  • C  πœ‹ 1 6 βˆ’ πœ‹ 𝑛 4 , πœ‹ 8 + πœ‹ 𝑛 2 ∢ 𝑛 ∈ β„€ 
  • D  πœ‹ 8 + πœ‹ 𝑛 8 , πœ‹ 4 βˆ’ πœ‹ 𝑛 4 ∢ 𝑛 ∈ β„€ 

Q12:

Find the value of ∠ πœƒ given c o s s i n ο€Ό 3 2 πœƒ  βˆ’ πœƒ = 0 where πœƒ is a positive acute angle.

Q13:

Find the value of πœƒ that satisfies s i n c o s ο€» 4 πœƒ βˆ’ πœ‹ 3  = 4 πœƒ where πœƒ ∈  0 , πœ‹ 2  .

  • A 5 πœ‹ 4 8
  • B 5 πœ‹ 2 4
  • C πœ‹ 4 8
  • D πœ‹ 2 4

Q14:

Find s i n c o s 3 πœƒ + 6 πœƒ given c o t t a n πœƒ = 2 πœƒ where πœƒ is a positive acute angle.

  • A0
  • B1
  • C βˆ’ 1 4
  • D 1 2

Q15:

Find π‘š ∠ 𝐡 given π‘š ∠ 𝐴 = 4 3 ∘ and s i n c o s 𝐡 = 𝐴 where 𝐡 is an acute angle. Give the answer to the nearest degree.

Q16:

Find the value of c s c c o t t a n c s c ( 9 0 + 𝛼 ) ( 2 7 0 + 𝛽 ) ( 2 7 0 βˆ’ 𝛼 ) ( 2 7 0 + 𝛽 ) ∘ ∘ ∘ ∘ given 1 7 𝛼 βˆ’ 8 = 0 s i n , where 0 < πœƒ < 9 0 ∘ ∘ , and 3 𝛽 + 4 = 0 t a n where 𝛽 is the largest angle in the range ] 0 , 3 6 0 [ ∘ ∘ .

  • A βˆ’ 8 5 1 8
  • B 8 5 1 8
  • C βˆ’ 5 4 4 4 0 5
  • D 5 4 4 4 0 5

Q17:

Find πœƒ in degrees given c s c s e c ( πœƒ βˆ’ 5 ) = ( 6 πœƒ βˆ’ 1 0 ) ∘ ∘ where πœƒ is a positive acute angle.

Q18:

Find πœƒ in degrees given c o s s i n ( πœƒ + 2 5 ) = ( 3 πœƒ + 5 ) ∘ ∘ where πœƒ is a positive acute angle.

Q19:

Find the value of s i n ( 9 0 βˆ’ πœƒ ) ∘ given s i n πœƒ = βˆ’ 3 5 where 1 8 0 ≀ πœƒ < 2 7 0 ∘ ∘ .

  • A βˆ’ 4 5
  • B βˆ’ 3 5
  • C 4 5
  • D 3 5

Q20:

Find the value of 𝑋 given c o s s i n ( 𝑋 + 6 ) = 5 1 ∘ ∘ where ( 𝑋 + 6 ) ∘ is an acute angle.

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

Q21:

Find the value of s i n ( 2 7 0 βˆ’ πœƒ ) ∘ given s i n πœƒ = 1 2 1 3 where 9 0 < πœƒ < 1 8 0 ∘ ∘ .

  • A 5 1 3
  • B βˆ’ 1 2 1 3
  • C βˆ’ 5 1 3
  • D 1 2 1 3

Q22:

Find the value of c o t ( 2 7 0 + πœƒ ) ∘ given t a n πœƒ = βˆ’ 4 3 where 9 0 < πœƒ < 1 8 0 ∘ ∘ .

  • A 4 3
  • B βˆ’ 4 3
  • C βˆ’ 3 4
  • D 3 4

Q23:

Find c o t ( 9 0 βˆ’ 3 πœƒ ) ∘ given t a n c o t πœƒ = 5 πœƒ where πœƒ is a positive acute angle.

  • A1
  • B √ 3 3
  • C √ 3
  • D √ 2 2

Q24:

Find the value of s i n t a n t a n ( 1 8 0 βˆ’ πœƒ ) + ( 9 0 βˆ’ πœƒ ) βˆ’ ( 2 7 0 βˆ’ πœƒ ) ∘ ∘ ∘ given c o s πœƒ = βˆ’ 4 5 where 9 0 < πœƒ < 1 8 0 ∘ ∘ .

  • A 3 5
  • B 4 9 1 5
  • C βˆ’ 4 9 1 5
  • D βˆ’ 3 5

Q25:

Find the value of c o s c o s ( 3 6 0 βˆ’ 𝐡 ) βˆ’ ( 9 0 βˆ’ 𝐡 ) ∘ ∘ given t a n 𝐡 = 4 3 where 0 < 𝐡 < πœ‹ 2 .

  • A βˆ’ 1 5
  • B0
  • C 1 5
  • D βˆ’ 6 5
  • E 6 5
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