Worksheet: Solving Equations Using Cofunction Identities

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In this worksheet, we will practice solving trigonometric equations using cofunction identities.

Q1:

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

  • A 1 7 8
  • B 8 1 7
  • C − 8 1 7
  • D − 1 7 8

Q2:

Find the value of c o t ( 2 7 0 + 𝜃 ) ∘ given t a n 𝜃 = − 4 3 where 9 0 < 𝜃 < 1 8 0 ∘ ∘ .

  • A 3 4
  • B − 4 3
  • C − 3 4
  • D 4 3

Q3:

Find the value of c o s ( 1 8 0 + 𝜃 ) ∘ given s i n ( 9 0 − 𝜃 ) = − 7 1 7 ∘ where 𝜃 is the smallest positive angle.

  • A 4 √ 1 5 1 7
  • B − 7 1 7
  • C − 4 √ 1 5 1 7
  • D 7 1 7

Q4:

Find the general solution of the equation s i n ( 9 0 − 𝜃 ) = 1 2 ∘ .

  • A ± 2 𝜋 3 + 2 𝜋 𝑛 where 𝑛 ∈ ℤ
  • B ± 𝜋 6 + 2 𝜋 𝑛 where 𝑛 ∈ ℤ
  • C ± 5 𝜋 6 + 2 𝜋 𝑛 where 𝑛 ∈ ℤ
  • D ± 𝜋 3 + 2 𝜋 𝑛 where 𝑛 ∈ ℤ

Q5:

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

  • A 𝜋 6 + 𝜋 𝑛 6 or 𝜋 4 − 𝜋 𝑛 4 , where 𝑛 ∈ ℤ
  • B 𝜋 3 + 𝜋 𝑛 3 or 𝜋 2 + 𝜋 𝑛 2 , where 𝑛 ∈ ℤ
  • C 𝜋 1 2 − 𝜋 𝑛 3 or 𝜋 8 + 𝜋 𝑛 2 , where 𝑛 ∈ ℤ
  • D 𝜋 1 2 + 𝜋 𝑛 3 or − 𝜋 8 − 𝜋 𝑛 2 , where 𝑛 ∈ ℤ

Q6:

Find the value of s i n t a n t a n ( 1 8 0 − 𝜃 ) + ( 9 0 − 𝜃 ) − ( 2 7 0 − 𝜃 ) ∘ ∘ ∘ given c o s 𝜃 = − 4 5 where 9 0 < 𝜃 < 1 8 0 ∘ ∘ .

  • A − 3 5
  • B 4 9 1 5
  • C − 4 9 1 5
  • D 3 5

Q7:

Find the value of s i n ( 9 0 − 𝜃 ) ∘ given s i n 𝜃 = − 3 5 where 1 8 0 ≤ 𝜃 < 2 7 0 ∘ ∘ .

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

Q8:

Find the value of s i n ( 2 7 0 − 𝜃 ) ∘ given s i n 𝜃 = 1 2 1 3 where 9 0 < 𝜃 < 1 8 0 ∘ ∘ .

  • A 1 2 1 3
  • B − 1 2 1 3
  • C − 5 1 3
  • D 5 1 3

Q9:

Find the value of c o t 𝜋 2 − 2 𝐵 given t a n 𝐵 = − 3 2 where 3 𝜋 2 < 𝐵 < 2 𝜋 .

  • A 5 6
  • B 6 5
  • C − 5 6
  • D 1 2 5

Q10:

Find s i n 8 𝜃 given s e c ( 9 0 + 𝜃 ) − 2 = 0 ∘ where 1 8 0 < 𝜃 < 2 7 0 ∘ ∘ .

  • A 1 2
  • B − 1 2
  • C − √ 3
  • D √ 3 2

Q11:

Find the value of c o t ( 𝜃 − 9 0 ) ∘ given s e c 𝜃 = − 1 7 1 5 where 9 0 < 𝜃 < 1 8 0 ∘ ∘ .

  • A 1 5 8
  • B − 8 1 5
  • C − 1 5 8
  • D 8 1 5

Q12:

Find the value of c o s c o s ( 3 6 0 − 𝐵 ) − ( 9 0 − 𝐵 ) ∘ ∘ given t a n 𝐵 = 4 3 where 0 < 𝐵 < 𝜋 2 .

  • A 6 5
  • B 1 5
  • C − 6 5
  • D − 1 5
  • E0

Q13:

Find 𝜃 in degrees given c s c s e c ( 𝜃 − 5 ) = ( 6 𝜃 − 1 0 ) ∘ ∘ where 𝜃 is a positive acute angle.

Q14:

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

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

Q15:

Find the value of c s c s e c c s c ( 5 6 ) ( 3 4 ) + ( 1 8 0 − 𝜃 ) ∘ ∘ ∘ given t a n c o t ( 𝜃 + 1 0 ) ( 𝜃 + 2 0 ) = 1 ∘ ∘ .

  • A − 2
  • B 1 2
  • C − 1 2
  • D2

Q16:

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

  • A 1 2
  • B1
  • C − 1 4
  • D0

Q17:

Find 𝑚 ∠ 𝐵 given 𝑚 ∠ 𝐴 = 4 3 ∘ and s i n c o s 𝐵 = 𝐴 where 𝐵 is an acute angle. Give the answer to the nearest degree.

Q18:

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 − 𝛼 ) ( 2 7 0 + 𝛽 ) ∘ ∘ ∘ ∘ given 1 7 𝛼 − 8 = 0 s i n , where 0 < 𝜃 < 9 0 ∘ ∘ , and 3 𝛽 + 4 = 0 t a n where 𝛽 is the largest angle in the range ] 0 , 3 6 0 [ ∘ ∘ .

  • A 5 4 4 4 0 5
  • B 8 5 1 8
  • C − 5 4 4 4 0 5
  • D − 8 5 1 8

Q19:

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

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

Q20:

Find c o t ( 9 0 − 3 𝜃 ) ∘ given t a n c o t 𝜃 = 5 𝜃 where 𝜃 is a positive acute angle.

  • A √ 2 2
  • B √ 3 3
  • C √ 3
  • D1

Q21:

Find the measure of ∠ 𝜃 given c s c s e c ( 𝜃 + 1 5 2 4 ′ ) = ( 𝜃 + 3 7 5 4 ′ ) ∘ ∘ 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 ′ ∘

Q22:

Find all possible solutions of 𝜃 that satisfy c o s s i n 2 𝜃 − 6 𝜃 = 0 .

  • A 𝜋 8 + 𝜋 𝑛 8 , 𝜋 4 − 𝜋 𝑛 4 ∶ 𝑛 ∈ ℤ
  • B 𝜋 4 + 𝜋 𝑛 4 , 𝜋 2 + 𝜋 𝑛 2 ∶ 𝑛 ∈ ℤ
  • C 𝜋 1 6 − 𝜋 𝑛 4 , 𝜋 8 + 𝜋 𝑛 2 ∶ 𝑛 ∈ ℤ
  • D 𝜋 1 6 + 𝜋 𝑛 4 , 𝜋 8 + 𝜋 𝑛 2 ∶ 𝑛 ∈ ℤ

Q23:

Find the value of ∠ 𝜃 given c o s s i n 3 2 𝜃 − 𝜃 = 0 where 𝜃 is a positive acute angle.

Q24:

Find the value of 𝜃 that satisfies s i n c o s 4 𝜃 − 𝜋 3 = 4 𝜃 where 𝜃 ∈ 0 , 𝜋 2 .

  • A 𝜋 2 4
  • B 5 𝜋 2 4
  • C 𝜋 4 8
  • D 5 𝜋 4 8

Q25:

Find the value of 𝜃 given t a n c o t 𝜃 − 𝜋 8 = 𝜃 where 𝜃 ∈ 0 , 𝜋 2 . Give the answer to the nearest second.

  • A 5 1 1 5 ′ 0 ′ ′ ∘
  • B 3 3 4 5 ′ 0 ′ ′ ∘
  • C 3 8 4 5 ′ 0 ′ ′ ∘
  • D 5 6 1 5 ′ 0 ′ ′ ∘

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