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Worksheet: Addition Formulae

Q1:

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

Q2:

Simplify s i n c o s c o s s i n 1 1 7 1 5 4 1 1 7 1 5 4 .

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

Q3:

Simplify s i n c o s c o s s i n 1 1 5 1 6 4 1 1 5 1 6 4 .

  • A c o s ( 4 9 )
  • B s i n ( 2 7 9 )
  • C c o s ( 2 7 9 )
  • D s i n ( 4 9 )

Q4:

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

  • A c o s ( 6 2 )
  • B s i n ( 1 4 2 )
  • C c o s ( 1 4 2 )
  • D s i n ( 6 2 )

Q5:

Simplify s i n c o s c o s s i n 1 3 2 2 7 1 3 2 2 7 .

  • A c o s 1 0 5
  • B s i n 1 5 9
  • C c o s 1 5 9
  • D s i n 1 0 5

Q6:

Simplify s i n c o s c o s s i n 5 6 1 4 1 5 6 1 4 1 .

  • A c o s ( 8 5 )
  • B s i n ( 1 9 7 )
  • C c o s ( 1 9 7 )
  • D s i n ( 8 5 )

Q7:

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

Q8:

Simplify c o s c o s s i n s i n 4 1 𝑋 7 𝑋 4 1 𝑋 7 𝑋 .

  • A s i n 3 4 𝑋
  • B c o s 3 4 𝑋
  • C s i n 4 8 𝑋
  • D c o s 4 8 𝑋

Q9:

Simplify c o s c o s s i n s i n ( 2 3 𝑋 + 2 5 𝑌 ) 2 5 𝑌 + ( 2 3 𝑋 + 2 5 𝑌 ) 2 5 𝑌 .

  • A s i n 2 3 𝑋
  • B c o s 2 5 𝑋
  • C s i n 2 5 𝑋
  • D c o s 2 3 𝑋
  • E c o s 4 8 𝑌

Q10:

Simplify s i n c o s c o s s i n ( 4 8 𝑋 + 4 2 𝑌 ) 4 2 𝑌 ( 4 8 𝑋 + 4 2 𝑌 ) 4 2 𝑌 .

  • A c o s 4 8 𝑋
  • B s i n 4 2 𝑋
  • C c o s 4 2 𝑋
  • D s i n 4 8 𝑋
  • E s i n 9 0 𝑌

Q11:

Using the relation t a n s i n c o s 𝜃 = 𝜃 𝜃 , 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

Q12:

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

Q13:

Simplify t a n t a n t a n t a n 1 5 9 1 1 4 1 + 1 5 9 1 1 4 .

  • A s i n 2 4 5
  • B t a n 2 7 3
  • C t a n 2 2 7 3
  • D t a n 4 5

Q14:

Simplify t a n t a n t a n t a n 1 5 6 4 3 1 + 1 5 6 4 3 .

  • A s i n 2 1 1 3
  • B t a n 1 9 9
  • C t a n 2 1 9 9
  • D t a n 1 1 3

Q15:

Simplify t a n t a n t a n t a n 1 6 7 1 0 2 1 + 1 6 7 1 0 2 .

  • A s i n 2 6 5
  • B t a n 2 6 9
  • C t a n 2 2 6 9
  • D t a n 6 5

Q16:

Simplify t a n t a n t a n t a n 2 2 1 6 1 + 2 2 1 6 .

  • A s i n 2 6
  • B t a n 3 8
  • C t a n 2 3 8
  • D t a n 6

Q17:

In the figure, which triangles are similar?

  • A 𝐵 𝐷 𝐸 , 𝐶 𝐷 𝐹 , and 𝐸 𝐵 𝐹
  • B 𝐷 𝐵 𝐶 , 𝐶 𝐷 𝐹 , 𝐴 𝐵 𝐶 , and 𝐸 𝐵 𝐹
  • C 𝐴 𝐷 𝐸 , 𝐸 𝐷 𝐵 , and 𝐴 𝐵 𝐶
  • D 𝐴 𝐷 𝐸 , 𝐹 𝐷 𝐶 , 𝐴 𝐵 𝐶 , and 𝐹 𝐵 𝐸
  • E 𝐷 𝐵 𝐶 , 𝐸 𝐷 𝐵 , and 𝐴 𝐵 𝐶

Given that 𝐵 𝐶 = 1 , find expressions for the lengths of 𝐴 𝐶 , 𝐶 𝐷 , 𝐴 𝐷 , and 𝐶 𝐹 .

  • A 𝐴 𝐶 = 𝛼 t a n , 𝐶 𝐷 = 𝛽 t a n , 𝐴 𝐷 = 𝛼 𝛽 t a n t a n , 𝐶 𝐹 = 𝛼 𝛽 t a n t a n
  • B 𝐴 𝐶 = 𝛼 t a n , 𝐶 𝐷 = 𝛽 t a n , 𝐴 𝐷 = 𝛼 𝛽 t a n t a n , 𝐶 𝐹 = 𝛽 𝛼 t a n t a n
  • C 𝐴 𝐶 = 1 𝛽 t a n , 𝐶 𝐷 = 1 𝛼 t a n , 𝐴 𝐷 = 1 𝛽 1 𝛼 t a n t a n , 𝐶 𝐹 = 𝛽 𝛼 t a n t a n
  • D 𝐴 𝐶 = 1 𝛼 t a n , 𝐶 𝐷 = 1 𝛽 t a n , 𝐴 𝐷 = 1 𝛼 1 𝛽 t a n t a n , 𝐶 𝐹 = 𝛽 𝛼 t a n t a n
  • E 𝐴 𝐶 = 𝛽 t a n , 𝐶 𝐷 = 𝛼 t a n , 𝐴 𝐷 = 𝛽 𝛼 t a n t a n , 𝐶 𝐹 = 𝛼 𝛽 t a n t a n

Find an expression for t a n ( 𝛼 𝛽 ) .

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

Q18:

Consider the given figure.

Find the lengths 𝐴 and 𝐵 in terms of 𝛼 and 𝛽 .

  • A 𝐴 = ( 𝛼 + 𝛽 ) c s c , 𝐵 = ( 𝛼 + 𝛽 ) s e c
  • B 𝐴 = ( 𝛼 + 𝛽 ) c o s , 𝐵 = ( 𝛼 + 𝛽 ) s i n
  • C 𝐴 = ( 𝛼 + 𝛽 ) s e c , 𝐵 = ( 𝛼 + 𝛽 ) c s c
  • D 𝐴 = ( 𝛼 + 𝛽 ) s i n , 𝐵 = ( 𝛼 + 𝛽 ) c o s
  • E 𝐴 = ( 𝛼 + 𝛽 ) t a n , 𝐵 = ( 𝛼 + 𝛽 ) c o t

Find the lengths 𝐶 , 𝐷 , 𝐸 , and 𝐹 in terms of 𝛼 and 𝛽 .

  • A 𝐶 = 𝛼 𝛽 s i n s i n , 𝐷 = 𝛼 𝛽 c o s s i n , 𝐸 = 𝛼 𝛽 s i n c o s , 𝐹 = 𝛼 𝛽 c o s c o s
  • B 𝐶 = 𝛼 𝛽 c o s s i n , 𝐷 = 𝛼 𝛽 s i n s i n , 𝐸 = 𝛽 𝛼 s i n s i n , 𝐹 = 𝛽 𝛼 s i n c o s
  • C 𝐶 = 𝛼 s i n , 𝐷 = 𝛼 c o s , 𝐸 = 𝛼 s i n , 𝐹 = 𝛼 c o s
  • D 𝐶 = 𝛼 𝛽 s i n s i n , 𝐷 = 𝛼 𝛽 c o s s i n , 𝐸 = 𝛽 𝛼 c o s s i n , 𝐹 = 𝛽 𝛼 c o s c o s

By writing 𝐴 and 𝐵 in terms of 𝐶 , 𝐷 , 𝐸 , and 𝐹 , find expressions for c o s ( 𝛼 + 𝛽 ) and s i n ( 𝛼 + 𝛽 ) in terms of c o s 𝛼 , c o s 𝛽 , s i n 𝛼 , and s i n 𝛽 .

  • A c o s c o s c o s s i n s i n ( 𝛼 + 𝛽 ) = 𝛼 𝛽 𝛼 𝛽 , s i n c o s s i n s i n c o s ( 𝛼 + 𝛽 ) = 𝛼 𝛽 + 𝛼 𝛽
  • B c o s c o s s i n ( 𝛼 + 𝛽 ) = 𝛼 𝛼 , s i n c o s s i n ( 𝛼 + 𝛽 ) = 𝛼 + 𝛼
  • C c o s c o s s i n c o s s i n ( 𝛼 + 𝛽 ) = 𝛼 𝛽 𝛼 𝛽 , s i n s i n s i n s i n s i n ( 𝛼 + 𝛽 ) = 𝛼 𝛽 + 𝛼 𝛽
  • D c o s c o s c o s s i n s i n ( 𝛼 + 𝛽 ) = 𝛼 𝛽 𝛼 𝛽 , s i n c o s s i n s i n c o s ( 𝛼 + 𝛽 ) = 𝛼 𝛽 + 𝛼 𝛽

Q19:

In the given figure, 𝑂 𝑀 𝑁 𝑇 is a rectangle and the length of 𝑂 𝑆 is 1.

Find the lengths of 𝑆 𝑇 and 𝑂 𝑇 in terms of 𝛼 and 𝛽 .

  • A 𝑆 𝑇 = ( 𝛼 𝛽 ) , 𝑂 𝑇 = ( 𝛼 𝛽 ) c s c s e c
  • B 𝑆 𝑇 = ( 𝛼 𝛽 ) , 𝑂 𝑇 = ( 𝛼 𝛽 ) c o s s i n
  • C 𝑆 𝑇 = ( 𝛼 𝛽 ) , 𝑂 𝑇 = ( 𝛼 𝛽 ) s e c c s c
  • D 𝑆 𝑇 = ( 𝛼 𝛽 ) , 𝑂 𝑇 = ( 𝛼 𝛽 ) s i n c o s
  • E 𝑆 𝑇 = ( 𝛼 𝛽 ) , 𝑂 𝑇 = ( 𝛼 𝛽 ) t a n c o t

Find 𝜃 in terms of 𝛼 and 𝛽 . Hence find the lengths of 𝑃 𝑄 and 𝑄 𝑆 .

  • A 𝜃 = 𝛼 , 𝑃 𝑄 = 𝛼 𝛽 , 𝑄 𝑆 = 𝛼 𝛽 c o s s i n s i n s i n
  • B 𝜃 = 9 0 𝛼 , 𝑃 𝑄 = 𝛼 𝛽 , 𝑄 𝑆 = 𝛼 𝛽 s i n s i n c o s s i n
  • C 𝜃 = 𝛼 , 𝑃 𝑄 = 𝛼 , 𝑄 𝑆 = 𝛼 c o s s i n
  • D 𝜃 = 𝛽 , 𝑃 𝑄 = 𝛽 𝛽 , 𝑄 𝑆 = 𝛽 c o s s i n s i n 2
  • E 𝜃 = 9 0 𝛼 , 𝑃 𝑄 = 𝛼 𝛽 , 𝑄 𝑆 = 𝛼 𝛽 c o s s i n s i n s i n

By considering a suitable angle, find the lengths of 𝑀 𝑃 and 𝑂 𝑀 .

  • A 𝑀 𝑃 = 𝛼 𝛽 , 𝑂 𝑀 = 𝛼 𝛽 c o s c o s s i n c o s
  • B 𝑀 𝑃 = 𝛽 , 𝑂 𝑀 = 𝛽 𝛽 c o s s i n c o s 2
  • C 𝑀 𝑃 = 𝛼 𝛽 , 𝑂 𝑀 = 𝛼 𝛽 c o s c o s s i n c o s
  • D 𝑀 𝑃 = 𝛼 , 𝑂 𝑀 = 𝛼 c o s s i n
  • E 𝑀 𝑃 = 𝛼 𝛽 , 𝑂 𝑀 = 𝛼 𝛽 s i n c o s c o s c o s

Use your answers to the previous parts of the question to find expressions for s i n ( 𝛼 𝛽 ) and c o s ( 𝛼 𝛽 ) .

  • A s i n s i n c o s c o s s i n c o s c o s c o s s i n s i n ( 𝛼 𝛽 ) = 𝛼 𝛽 𝛼 𝛽 , ( 𝛼 𝛽 ) = 𝛼 𝛽 + 𝛼 𝛽
  • B s i n c o s c o s s i n s i n c o s s i n c o s c o s s i n ( 𝛼 𝛽 ) = 𝛼 𝛽 𝛼 𝛽 , ( 𝛼 𝛽 ) = 𝛼 𝛽 + 𝛼 𝛽
  • C s i n s i n s i n c o s c o s c o s c o s s i n s i n c o s ( 𝛼 𝛽 ) = 𝛼 𝛽 + 𝛼 𝛽 , ( 𝛼 𝛽 ) = 𝛼 𝛽 𝛼 𝛽
  • D s i n s i n s i n c o s c o s c o s ( 𝛼 𝛽 ) = 𝛼 𝛽 , ( 𝛼 𝛽 ) = 𝛼 𝛽
  • E s i n s i n c o s c o s c o s s i n ( 𝛼 𝛽 ) = 𝛼 𝛼 , ( 𝛼 𝛽 ) = 𝛼 + 𝛼