Worksheet: Applications of Trigonometric Addition Formula

In this worksheet, we will practice using addition formulas to simplify trigonometric expressions.

Q1:

Given that c o s 𝜃 = 5 3 , where 0 𝜃 𝜋 , and c o s 𝜑 = 2 3 , where 0 𝜑 𝜋 , find the exact value of c o s ( 𝜑 + 𝜃 ) .

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

Q2:

Simplify s i n c o s c o s s i n 1 4 7 1 2 0 1 4 7 1 2 0 .

  • A c o s 2 7
  • B s i n 2 6 7
  • C c o s 2 6 7
  • D s i n 2 7

Q3:

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

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

Q4:

Find t a n ( 𝐴 + 𝐵 ) , given t a n 𝐴 = 4 3 where 3 𝜋 2 < 𝐴 < 2 𝜋 and t a n 𝐵 = 1 5 8 where 0 < 𝐵 < 𝜋 2 .

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

Q5:

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

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

Q6:

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

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

Q7:

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

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

Q8:

Find c o t ( 𝐴 𝐵 ) given c o s 𝐴 = 4 5 and s i n 𝐵 = 7 2 5 where 𝐴 and 𝐵 are two positive acute angles.

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

Q9:

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

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

Q10:

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 𝐵 = 5 1 2 , where 9 0 < 𝐵 < 1 8 0 , find c o t ( 𝐴 + 𝐵 ) .

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

Q11:

Find c o s ( 𝐴 + 𝐵 ) given t a n 𝐴 = 5 1 2 where 𝜋 < 𝐴 < 3 𝜋 2 and t a n 𝐵 = 4 3 where 𝜋 2 < 𝐵 < 𝜋 .

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

Q12:

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

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

Q13:

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

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

Q14:

Given that s i n 𝜃 = 3 2 , where 0 𝜃 𝜋 2 , and c o s 𝜑 = 2 2 3 , where 3 𝜋 2 𝜑 2 𝜋 , find the exact value of t a n ( 𝜑 𝜃 ) .

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

Q15:

Given that c o s 𝜃 = 3 5 , where 𝜋 2 𝜃 𝜋 , and s i n 𝜑 = 1 3 , where 𝜋 2 𝜑 𝜋 , find the exact value of s i n ( 𝜑 𝜃 ) .

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

Q16:

Given that c o s 𝜃 = 3 4 , where 𝜋 2 𝜃 𝜋 , and c o s 𝜑 = 2 2 , where 𝜋 𝜑 3 𝜋 2 , find the exact value of t a n ( 𝜑 + 𝜃 ) .

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

Q17:

Given that s i n 𝜃 = 3 3 , where 𝜋 𝜃 3 𝜋 2 , and s i n 𝜑 = 1 3 , where 𝜋 2 𝜑 𝜋 , find the exact value of s i n ( 𝜑 + 𝜃 ) .

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

Q18:

Simplify c o s c o s s i n s i n 2 𝑋 2 2 𝑋 2 𝑋 2 2 𝑋 .

  • A s i n 2 0 𝑋
  • B c o s 2 0 𝑋
  • C s i n 2 4 𝑋
  • D c o s 2 4 𝑋

Q19:

Using the relation t a n t a n t a n t a n t a n ( 𝛼 + 𝛽 ) = 𝛼 + 𝛽 1 𝛼 𝛽 , find an expression for t a n ( 𝛼 𝛽 ) in terms of t a n 𝛼 and t a n 𝛽 which holds when ( 𝛼 𝛽 ) 𝜋 2 + 𝜋 𝑛 .

  • A t a n t a n t a n t a n t a n ( 𝛼 𝛽 ) = 𝛼 + 𝛽 1 𝛼 𝛽
  • B t a n t a n t a n t a n t a n ( 𝛼 𝛽 ) = 𝛼 + 𝛽 1 + 𝛼 𝛽
  • C t a n t a n t a n t a n t a n ( 𝛼 𝛽 ) = 𝛼 𝛽 1 𝛼 𝛽
  • D t a n t a n t a n t a n t a n ( 𝛼 𝛽 ) = 𝛼 𝛽 1 + 𝛼 𝛽
  • E t a n t a n t a n t a n t a n ( 𝛼 𝛽 ) = 𝛼 𝛽 𝛼 + 𝛽

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