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Worksheet: Product-to-Sum Identities

In this worksheet, we will practice using the product-to-sum identity to evaluate a trigonometric expression.

Q1:

Simplify s i n c o s c o s s i n ( π‘₯ + 𝑦 ) 𝑦 βˆ’ ( π‘₯ + 𝑦 ) 𝑦 .

  • A 0
  • B c o s π‘₯
  • C 2 π‘₯ 𝑦 s i n c o s 2
  • D s i n π‘₯

Q2:

Simplify c o s c o s ( 𝐴 βˆ’ 𝐡 ) βˆ’ ( 𝐴 + 𝐡 ) .

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

Q3:

Find the value of , given where .

  • A
  • B
  • C
  • D

Q4:

Find the value of s i n s i n s i n s i n s i n s i n 2 2 2 2 2 2 πœ‹ 1 2 + 3 πœ‹ 1 2 + 5 πœ‹ 1 2 + 7 πœ‹ 1 2 + 9 πœ‹ 1 2 + 1 1 πœ‹ 1 2 .

Q5:

The intensity of an electric current is given by 𝐢 = 5 2 ( 7 5 𝑑 ) s i n ∘ where 𝑑 is the time in seconds. Find the intensity after one second without using a calculator using sum and product formulae.

  • A 5 ο€» √ 6 + √ 2 
  • B 5 ο€» √ 6 βˆ’ √ 2  8
  • C 5 ο€» βˆ’ √ 6 βˆ’ √ 2 
  • D 5 ο€» √ 6 + √ 2  8

Q6:

Find the exact value of 2 ( 1 9 5 ) ( 4 5 ) c o s s i n .

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

Q7:

Find, without using a calculator, the value of given where and where .

  • A
  • B
  • C
  • D
  • E

Q8:

Write the expression s i n s i n ( 8 0 ) + ( 2 0 ) as a product of trigonometric expressions.

  • A 2 ( 5 0 ) ( 3 0 ) c o s c o s
  • B s i n c o s ( 5 0 ) ( 3 0 )
  • C 2 ( 5 0 ) ( 3 0 ) s i n s i n
  • D 2 ( 5 0 ) ( 3 0 ) s i n c o s
  • E s i n s i n ( 5 0 ) ( 3 0 )

Q9:

Find the value of c o s c o s s i n s i n 5 𝑋 𝑋 + 5 𝑋 𝑋 given s i n 2 𝑋 = 2 1 1 .

  • A 1 2 9 1 2 1
  • B 1 1 7 1 2 1
  • C 1 2 5 1 2 1
  • D 1 1 3 1 2 1

Q10:

Find, without using a calculator, the value of given where and where .

  • A
  • B
  • C
  • D8
  • E

Q11:

Write the expression 4 ( 7 0 ) ( 3 3 ) c o s c o s as a sum or difference of trigonometric expressions.

  • A 2 ( 1 0 3 ) βˆ’ 2 ( 3 7 ) c o s c o s
  • B c o s c o s ( 3 7 ) + ( 1 0 3 )
  • C 2 3 7 + 2 1 0 3 s i n s i n
  • D 2 ( 3 7 ) + 2 ( 1 0 3 ) c o s c o s
  • E 2 1 0 3 βˆ’ 2 3 7 s i n s i n

Q12:

Find the value of s i n s i n s i n s i n s i n s i n s i n s i n 2 2 2 2 2 2 2 2 πœ‹ 1 6 + 3 πœ‹ 1 6 + 5 πœ‹ 1 6 + 7 πœ‹ 1 6 + 9 πœ‹ 1 6 + 1 1 πœ‹ 1 6 + 1 3 πœ‹ 1 6 + 1 5 πœ‹ 1 6 .

Q13:

Find c o s ( 2 π‘Ž + 𝑏 ) given 1 2 π‘Ž = 5 t a n and 4 𝑏 = 3 t a n where π‘Ž and 𝑏 are acute angles.

  • A βˆ’ 5 8 8 4 5
  • B 8 3 6 8 4 5
  • C βˆ’ 3 3 6 5
  • D 1 1 6 8 4 5

Q14:

Find the value of t a n ( 𝐴 βˆ’ 2 𝐡 ) given 2 9 𝐴 + 2 1 = 0 c o s where 𝐴 is the smallest positive angle and 3 𝐡 βˆ’ 4 = 0 t a n where 1 8 0 < 𝐡 < 2 7 0 ∘ ∘ .

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

Q15:

Simplify s i n s i n ( 𝐴 + 𝐡 ) βˆ’ ( 𝐴 βˆ’ 𝐡 ) .

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

Q16:

𝐴 𝐡 𝐢 is a triangle where t a n 𝐴 = 1 2 and 𝐡 = 2 𝐴 . Find c o s 𝐢 without using a calculator.

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

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