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Worksheet: Applications of Trigonometric Addition Formula

Q1:

Given that c o s πœƒ = βˆ’ √ 5 3 , where 0 ≀ πœƒ ≀ πœ‹ , and c o s πœ‘ = √ 2 3 , where 0 ≀ πœ‘ ≀ πœ‹ , find the exact value of c o s ( πœ‘ + πœƒ ) .

  • A 2 √ 7 + √ 1 0 9
  • B βˆ’ √ 1 4 + √ 1 0 9
  • C √ 1 4 + √ 1 0 9
  • D βˆ’ 2 √ 7 + √ 1 0 9
  • E 2 √ 7 + √ 1 0 3

Q2:

Find c o t ( 𝐴 + 𝐡 ) given s i n 𝐴 = 5 1 3 and t a n 𝐡 = 3 4 where 𝐴 and 𝐡 are acute angles.

  • A βˆ’ 5 6 3 3
  • B 5 6 3 3
  • C βˆ’ 8 9
  • D 3 3 5 6
  • E 8 9

Q3:

Given that s i n πœƒ = √ 3 2 , where 0 ≀ πœƒ ≀ πœ‹ 2 , and c o s πœ‘ = 2 √ 2 3 , where 3 πœ‹ 2 ≀ πœ‘ ≀ 2 πœ‹ , find the exact value of t a n ( πœ‘ βˆ’ πœƒ ) .

  • A 7 √ 3 + 4 √ 2 2 3
  • B 9 √ 3 βˆ’ 8 √ 2 5
  • C 8 √ 2 + 9 √ 3 2 3
  • D βˆ’ 8 √ 2 + 9 √ 3 5
  • E 1 1 βˆ’ 4 √ 6 5

Q4:

Find s i n ( 𝐴 + 𝐡 ) given s i n 𝐴 = βˆ’ 2 4 2 5 where 2 7 0 ≀ 𝐴 < 3 6 0 ∘ ∘ and c o s 𝐡 = 4 5 where 0 ≀ 𝐡 < 9 0 ∘ ∘ .

  • A βˆ’ 1 1 7 1 2 5
  • B βˆ’ 3 5
  • C 3 5
  • D βˆ’ 5 3
  • E 1 1 7 1 2 5

Q5:

Find c o s ( 𝐴 βˆ’ 𝐡 ) given s i n s i n 𝐴 𝐡 = 6 1 3 and c o s c o s 𝐴 𝐡 = 9 2 3 where 𝐴 and 𝐡 are acute angles.

  • A 2 1 2 9 9
  • B βˆ’ 2 1 2 9 9
  • C βˆ’ 2 5 5 2 9 9
  • D 2 5 5 2 9 9

Q6:

Find c o t ( 𝐴 βˆ’ 𝐡 ) given c o s 𝐴 = 4 5 and s i n 𝐡 = 7 2 5 where 𝐴 and 𝐡 are two positive acute angles.

  • A βˆ’ 4 3
  • B 4 4 1 1 7
  • C βˆ’ 1 0 0 1 1 7
  • D 1 1 7 4 4
  • E 4 3

Q7:

Find t a n ( 𝐴 + 𝐡 ) , given t a n 𝐴 = βˆ’ 4 3 where 3 πœ‹ 2 < 𝐴 < 2 πœ‹ and t a n 𝐡 = 1 5 8 where 0 < 𝐡 < πœ‹ 2 .

  • A 7 7 3 6
  • B 8 4 1 3
  • C βˆ’ 1 1 1 2
  • D 1 3 8 4
  • E 1 1 1 2

Q8:

Find c o s ( 𝐴 + 𝐡 ) given t a n 𝐴 = 5 1 2 where πœ‹ < 𝐴 < 3 πœ‹ 2 and t a n 𝐡 = βˆ’ 4 3 where πœ‹ 2 < 𝐡 < πœ‹ .

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

Q9:

Find c o s ( 𝐴 + 𝐡 ) given t a n 𝐴 = βˆ’ 7 2 4 where 3 πœ‹ 2 < 𝐴 < 2 πœ‹ and t a n 𝐡 = 8 1 5 where 0 < 𝐡 < πœ‹ 2 .

  • A 3 0 4 4 2 5
  • B 4 2 5 4 1 6
  • C βˆ’ 4 1 6 4 2 5
  • D 4 1 6 4 2 5
  • E βˆ’ 3 0 4 4 2 5

Q10:

Given that s i n πœƒ = βˆ’ √ 3 3 , where πœ‹ ≀ πœƒ ≀ 3 πœ‹ 2 , and s i n πœ‘ = 1 3 , where πœ‹ 2 ≀ πœ‘ ≀ πœ‹ , find the exact value of s i n ( πœ‘ + πœƒ ) .

  • A √ 6 3
  • B 5 √ 3 9
  • C 5 √ 3 3
  • D √ 6 9
  • E √ 3 3

Q11:

Find s e c ( 𝐴 βˆ’ 𝐡 ) without using a calculator given s e c 𝐴 = 5 4 and c s c 𝐡 = 1 3 1 2 where 𝐴 and 𝐡 are acute angles.

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

Q12:

Given that c o s πœƒ = βˆ’ 3 4 , where πœ‹ 2 ≀ πœƒ ≀ πœ‹ , and c o s πœ‘ = βˆ’ √ 2 2 , where πœ‹ ≀ πœ‘ ≀ 3 πœ‹ 2 , find the exact value of t a n ( πœ‘ + πœƒ ) .

  • A βˆ’ 8 + 3 √ 7
  • B 8 + 3 √ 7
  • C βˆ’ 1 + √ 3 2
  • D 3 βˆ’ √ 7 3 + √ 7
  • E 3 + √ 7 3 βˆ’ √ 7

Q13:

Given that s i n 2 𝐴 = 4 4 1 8 4 1 , where 1 8 0 < 𝐴 < 2 7 0 ∘ ∘ , and t a n 𝐡 = βˆ’ 5 1 2 , where 9 0 < 𝐡 < 1 8 0 ∘ ∘ , find c o t ( 𝐴 + 𝐡 ) .

  • A 1 3 5 1 5 2
  • B 1 5 2 3 4 5
  • C βˆ’ 1 3 5 1 5 2
  • D 3 4 5 1 5 2

Q14:

Find c o s ( 𝐴 + 𝐡 ) , given s i n s i n 𝐴 𝐡 = 5 2 9 and c o s c o s 𝐴 𝐡 = 1 3 where 𝐴 and 𝐡 are acute angles.

  • A βˆ’ 1 4 8 7
  • B 4 4 8 7
  • C βˆ’ 4 4 8 7
  • D 1 4 8 7

Q15:

Find c o s ( 𝐴 + 𝐡 ) , given s i n s i n 𝐴 𝐡 = 2 3 and c o s c o s 𝐴 𝐡 = 1 8 1 9 where 𝐴 and 𝐡 are acute angles.

  • A βˆ’ 1 6 5 7
  • B 9 2 5 7
  • C βˆ’ 9 2 5 7
  • D 1 6 5 7

Q16:

Given that s i n 𝐴 = 4 5 , where 9 0 < 𝐴 < 1 8 0 ∘ ∘ , and t a n 𝐡 = 1 2 5 , where 1 8 0 < 𝐡 < 2 7 0 ∘ ∘ , determine s i n ( 𝐴 + 𝐡 ) .

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

Q17:

Given that s i n 𝐴 = βˆ’ 8 1 7 , where 1 8 0 < 𝐴 < 2 7 0 ∘ ∘ , and t a n 𝐡 = βˆ’ 4 3 , where 2 7 0 < 𝐡 < 3 6 0 ∘ ∘ , determine s i n ( 𝐴 + 𝐡 ) .

  • A 8 4 8 5
  • B 8 5 3 6
  • C βˆ’ 8 4 8 5
  • D 3 6 8 5

Q18:

Given that s i n 𝐴 = 4 5 , where 0 < 𝐴 < 9 0 ∘ ∘ and t a n 𝐡 = 7 2 4 , where 1 8 0 < 𝐡 < 2 7 0 ∘ ∘ , determine c o s ( 𝐴 + 𝐡 ) .

  • A βˆ’ 4 5
  • B βˆ’ 1 2 5 4 4
  • C 4 5
  • D βˆ’ 4 4 1 2 5

Q19:

Given that c o s πœƒ = βˆ’ 3 5 , where πœ‹ 2 ≀ πœƒ ≀ πœ‹ , and s i n πœ‘ = 1 3 , where πœ‹ 2 ≀ πœ‘ ≀ πœ‹ , find the exact value of s i n ( πœ‘ βˆ’ πœƒ ) .

  • A 6 √ 2 βˆ’ 4 1 5
  • B 4 βˆ’ 6 √ 2 1 5
  • C 8 √ 2 + 3 1 5
  • D 8 √ 2 βˆ’ 3 1 5
  • E 6 √ 2 + 4 1 5