Worksheet: Product-to-Sum Identities

In this worksheet, we will practice using the product-to-sum identity to simplify and evaluating trigonometric expression.

Q1:

Simplify sincoscossin(𝑥+𝑦)𝑦(𝑥+𝑦)𝑦.

  • A 2 𝑥 𝑦 s i n c o s
  • B c o s 𝑥
  • C0
  • D s i n 𝑥

Q2:

Simplify coscos(𝐴𝐵)(𝐴+𝐵).

  • A0
  • 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 sinsin(𝐴+𝐵)(𝐴𝐵).

  • A0
  • B 2 𝐴 𝐵 s i n c o s
  • C 2 𝐴 𝐵 s i n s i n
  • D 2 𝐴 𝐵 c o s s i n

Q4:

Write the expression 4(70)(33)coscos as a sum or difference of trigonometric expressions.

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

Q5:

Find the exact value of 2(195)(45)cossin.

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

Q6:

Write the expression sinsin(80)+(20) as a product of trigonometric expressions.

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

Q7:

Find the value of coscossinsin5𝑋𝑋+5𝑋𝑋 given sin2𝑋=38.

  • A 2 3 3 2
  • B 4 1 3 2
  • C 7 3 6 4
  • D 5 5 6 4

Q8:

Find cos(2𝑎+𝑏) given 4𝑎=3tan and 3𝑏=4tan where 𝑎 and 𝑏 are acute angles.

  • A0
  • B 3 5
  • C 1 1 7 1 2 5
  • D 3 1 0

Q9:

The intensity of an electric current is given by 𝐶=52(75𝑡)sin 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

Q10:

Find the value of sinsinsinsinsinsin𝜋12+3𝜋12+5𝜋12+7𝜋12+9𝜋12+11𝜋12.

Q11:

Find the value of sinsinsinsinsinsinsinsin𝜋16+3𝜋16+5𝜋16+7𝜋16+9𝜋16+11𝜋16+13𝜋16+15𝜋16.

Q12:

Expand sin𝜃+𝜋6 using sum and difference formulas.

  • A 2 𝜃 + 3 𝜃 s i n c o s
  • B 1 2 𝜃 + 3 𝜃 s i n c o s
  • C 1 2 3 𝜃 + 𝜃 s i n c o s
  • D 1 2 ( 𝜃 + 𝜃 ) s i n c o s

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