Worksheet: Product-to-Sum Identities

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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 ( 𝑥 + 𝑦 ) 𝑦 − ( 𝑥 + 𝑦 ) 𝑦 .

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

Q2:

Simplify c o s c o s ( 𝐴 − 𝐵 ) − ( 𝐴 + 𝐵 ) .

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

Q3:

Simplify s i n s i n ( 𝐴 + 𝐵 ) − ( 𝐴 − 𝐵 ) .

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

Q4:

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 ) − 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 − 2 3 7 s i n s i n

Q5:

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

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

Q6:

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 )

Q7:

Find the value of c o s 3 𝑋 , given 5 𝑋 + 4 = 0 c o s where 2 𝑋 ∈ ] 0 , 2 𝜋 [ .

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

Q8:

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

Q9:

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

  • A − 5 8 8 4 5
  • B 8 3 6 8 4 5
  • C − 3 3 6 5
  • D 1 1 6 8 4 5

Q10:

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

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

Q11:

Find, without using a calculator, the value of t a n 2 ( 𝐵 + 𝐴 ) given t a n 𝐴 = 5 7 where 𝐴 ∈ 0 , 𝜋 2 and t a n 𝐵 = − 3 7 where 𝐵 ∈ 𝜋 2 , 𝜋 .

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

Q12:

Find, without using a calculator, the value of t a n ( 2 𝐴 + 𝐵 ) given t a n 𝐴 = 3 where 𝐴 ∈ 0 , 𝜋 2 and t a n 𝐵 = − 7 4 where 𝐵 ∈ 𝜋 2 , 𝜋 .

  • A 1 5
  • B − 8
  • C − 1 5
  • D8
  • E 5 1 2

Q13:

Find the value of t a n ( 𝐴 − 2 𝐵 ) given 2 9 𝐴 + 2 1 = 0 c o s where 𝐴 is the smallest positive angle and 3 𝐵 − 4 = 0 t a n where 1 8 0 < 𝐵 < 2 7 0 ∘ ∘ .

  • A 6 2 7 3 6 4
  • B − 3 6 4 6 2 7
  • C − 6 2 7 3 6 4
  • D 3 6 4 6 2 7

Q14:

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

  • A 5 √ 6 + √ 2
  • B 5 √ 6 − √ 2 8
  • C 5 − √ 6 − √ 2
  • D 5 √ 6 + √ 2 8

Q15:

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

Q16:

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 1 6 + 3 𝜋 1 6 + 5 𝜋 1 6 + 7 𝜋 1 6 + 9 𝜋 1 6 + 1 1 𝜋 1 6 + 1 3 𝜋 1 6 + 1 5 𝜋 1 6 .

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