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Lesson: Addition Formulae

Worksheet • 19 Questions

Q1:

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

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

Q2:

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

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

Q3:

Using the relation t a n t a n t a n t a n t a n ( 𝛼 + 𝛽 ) = 𝛼 + 𝛽 1 βˆ’ 𝛼 𝛽 , find an expression for t a n ( 𝛼 βˆ’ 𝛽 ) in terms of t a n 𝛼 and t a n 𝛽 which holds when ( 𝛼 βˆ’ 𝛽 ) β‰  πœ‹ 2 + πœ‹ 𝑛 .

  • A t a n t a n t a n t a n t a n ( 𝛼 βˆ’ 𝛽 ) = 𝛼 βˆ’ 𝛽 1 + 𝛼 𝛽
  • B t a n t a n t a n t a n t a n ( 𝛼 βˆ’ 𝛽 ) = 𝛼 βˆ’ 𝛽 𝛼 + 𝛽
  • C t a n t a n t a n t a n t a n ( 𝛼 βˆ’ 𝛽 ) = 𝛼 + 𝛽 1 + 𝛼 𝛽
  • D t a n t a n t a n t a n t a n ( 𝛼 βˆ’ 𝛽 ) = 𝛼 βˆ’ 𝛽 1 βˆ’ 𝛼 𝛽
  • E t a n t a n t a n t a n t a n ( 𝛼 βˆ’ 𝛽 ) = 𝛼 + 𝛽 1 βˆ’ 𝛼 𝛽

Q4:

Simplify s i n c o s c o s s i n ( 4 8 𝑋 + 4 2 π‘Œ ) 4 2 π‘Œ βˆ’ ( 4 8 𝑋 + 4 2 π‘Œ ) 4 2 π‘Œ .

  • A s i n 4 8 𝑋
  • B s i n 9 0 π‘Œ
  • C s i n 4 2 𝑋
  • D c o s 4 2 𝑋
  • E c o s 4 8 𝑋

Q5:

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

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

Q6:

Simplify s i n c o s c o s s i n 1 1 7 1 5 4 βˆ’ 1 1 7 1 5 4 ∘ ∘ ∘ ∘ .

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

Q7:

Simplify s i n c o s c o s s i n 1 1 5 1 6 4 βˆ’ 1 1 5 1 6 4 ∘ ∘ ∘ ∘ .

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

Q8:

Simplify s i n c o s c o s s i n 4 0 1 0 2 βˆ’ 4 0 1 0 2 ∘ ∘ ∘ ∘ .

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

Q9:

Simplify s i n c o s c o s s i n 1 3 2 2 7 βˆ’ 1 3 2 2 7 ∘ ∘ ∘ ∘ .

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

Q10:

Simplify s i n c o s c o s s i n 5 6 1 4 1 βˆ’ 5 6 1 4 1 ∘ ∘ ∘ ∘ .

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

Q11:

Simplify c o s c o s s i n s i n ( 2 3 𝑋 + 2 5 π‘Œ ) 2 5 π‘Œ + ( 2 3 𝑋 + 2 5 π‘Œ ) 2 5 π‘Œ .

  • A c o s 2 3 𝑋
  • B c o s 4 8 π‘Œ
  • C c o s 2 5 𝑋
  • D s i n 2 5 𝑋
  • E s i n 2 3 𝑋

Q12:

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 βˆ’ 𝛽 ) , 𝑂 𝑇 = ( 𝛼 βˆ’ 𝛽 ) s i n c o s
  • B 𝑆 𝑇 = ( 𝛼 βˆ’ 𝛽 ) , 𝑂 𝑇 = ( 𝛼 βˆ’ 𝛽 ) t a n c o t
  • C 𝑆 𝑇 = ( 𝛼 βˆ’ 𝛽 ) , 𝑂 𝑇 = ( 𝛼 βˆ’ 𝛽 ) c o s s i n
  • D 𝑆 𝑇 = ( 𝛼 βˆ’ 𝛽 ) , 𝑂 𝑇 = ( 𝛼 βˆ’ 𝛽 ) s e c c s c
  • E 𝑆 𝑇 = ( 𝛼 βˆ’ 𝛽 ) , 𝑂 𝑇 = ( 𝛼 βˆ’ 𝛽 ) c s c s e c

Find πœƒ in terms of 𝛼 and 𝛽 . Hence find the lengths of 𝑃 𝑄 and 𝑄 𝑆 .

  • A πœƒ = 𝛼 , 𝑃 𝑄 = 𝛼 𝛽 , 𝑄 𝑆 = 𝛼 𝛽 c o s s i n s i n s i n
  • B πœƒ = 9 0 βˆ’ 𝛼 , 𝑃 𝑄 = 𝛼 𝛽 , 𝑄 𝑆 = 𝛼 𝛽 ∘ c o s s i n s i n s i n
  • C πœƒ = 𝛽 , 𝑃 𝑄 = 𝛽 𝛽 , 𝑄 𝑆 = 𝛽 c o s s i n s i n 2
  • D πœƒ = 9 0 βˆ’ 𝛼 , 𝑃 𝑄 = 𝛼 𝛽 , 𝑄 𝑆 = 𝛼 𝛽 ∘ s i n s i n c o s s i n
  • E πœƒ = 𝛼 , 𝑃 𝑄 = 𝛼 , 𝑄 𝑆 = 𝛼 c o s 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 = 𝛼 𝛽 , 𝑂 𝑀 = 𝛼 𝛽 c o s c o s s i n c o s
  • C 𝑀 𝑃 = 𝛽 , 𝑂 𝑀 = 𝛽 𝛽 c o s s i n c o s 2
  • D 𝑀 𝑃 = 𝛼 𝛽 , 𝑂 𝑀 = 𝛼 𝛽 s i n c o s c o s c o s
  • E 𝑀 𝑃 = 𝛼 , 𝑂 𝑀 = 𝛼 c o s s i n

Use your answers to the previous parts of the question to find expressions for s i n ( 𝛼 βˆ’ 𝛽 ) and c o s ( 𝛼 βˆ’ 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 βˆ’ 𝛽 ) = 𝛼 𝛽 βˆ’ 𝛼 𝛽 , ( 𝛼 βˆ’ 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 βˆ’ 𝛽 ) = 𝛼 𝛽 βˆ’ 𝛼 𝛽 , ( 𝛼 βˆ’ 𝛽 ) = 𝛼 𝛽 + 𝛼 𝛽
  • C s i n s i n s i n c o s c o s c o s ( 𝛼 βˆ’ 𝛽 ) = 𝛼 βˆ’ 𝛽 , ( 𝛼 βˆ’ 𝛽 ) = 𝛼 βˆ’ 𝛽
  • D 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 βˆ’ 𝛽 ) = 𝛼 𝛽 + 𝛼 𝛽 , ( 𝛼 βˆ’ 𝛽 ) = 𝛼 𝛽 βˆ’ 𝛼 𝛽
  • E s i n s i n c o s c o s c o s s i n ( 𝛼 βˆ’ 𝛽 ) = 𝛼 βˆ’ 𝛼 , ( 𝛼 βˆ’ 𝛽 ) = 𝛼 + 𝛼

Q13:

In the figure, which triangles are similar?

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

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

  • A 𝐴 𝐢 = 𝛼 t a n , 𝐢 𝐷 = 𝛽 t a n , 𝐴 𝐷 = 𝛼 βˆ’ 𝛽 t a n t a n , 𝐢 𝐹 = 𝛼 𝛽 t a n t a n
  • B 𝐴 𝐢 = 𝛽 t a n , 𝐢 𝐷 = 𝛼 t a n , 𝐴 𝐷 = 𝛽 βˆ’ 𝛼 t a n t a n , 𝐢 𝐹 = 𝛼 𝛽 t a n t a n
  • C 𝐴 𝐢 = 1 𝛼 t a n , 𝐢 𝐷 = 1 𝛽 t a n , 𝐴 𝐷 = 1 𝛼 βˆ’ 1 𝛽 t a n t a n , 𝐢 𝐹 = 𝛽 𝛼 t a n t a n
  • D 𝐴 𝐢 = 𝛼 t a n , 𝐢 𝐷 = 𝛽 t a n , 𝐴 𝐷 = 𝛼 βˆ’ 𝛽 t a n t a n , 𝐢 𝐹 = 𝛽 𝛼 t a n t a n
  • E 𝐴 𝐢 = 1 𝛽 t a n , 𝐢 𝐷 = 1 𝛼 t a n , 𝐴 𝐷 = 1 𝛽 βˆ’ 1 𝛼 t a n t a n , 𝐢 𝐹 = 𝛽 𝛼 t a n t a n

Find an expression for t a n ( 𝛼 βˆ’ 𝛽 ) .

  • A t a n t a n t a n t a n t a n ( 𝛼 βˆ’ 𝛽 ) = 𝐸 𝐷 𝐸 𝐡 = 𝐴 𝐷 𝐡 𝐹 = 𝛼 βˆ’ 𝛽 1 + 𝛼 𝛽
  • B t a n t a n t a n ( 𝛼 βˆ’ 𝛽 ) = 𝐸 𝐷 𝐸 𝐡 = 𝐴 𝐷 𝐡 𝐹 = βˆ’ 1 𝛼 + 𝛽 t a n t a n  
  • C t a n t a n t a n t a n t a n ( 𝛼 βˆ’ 𝛽 ) = 𝐸 𝐷 𝐸 𝐡 = 𝐴 𝐷 𝐡 𝐹 = 𝛼 βˆ’ 𝛽 𝛼 𝛽
  • D t a n t a n t a n t a n t a n ( 𝛼 βˆ’ 𝛽 ) = 𝐸 𝐷 𝐸 𝐡 = 𝐴 𝐷 𝐡 𝐹 = 𝛽 βˆ’ 𝛼 1 + 𝛼 𝛽
  • E t a n t a n t a n ( 𝛼 βˆ’ 𝛽 ) = 𝐸 𝐷 𝐸 𝐡 = 𝐴 𝐷 𝐡 𝐹 = 1 βˆ’ 𝛼 + 𝛽 t a n t a n  

Q14:

Consider the given figure.

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

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

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

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

By writing 𝐴 and 𝐡 in terms of 𝐢 , 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 βˆ’ 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 βˆ’ 𝛼 , s i n c o s s i n ( 𝛼 + 𝛽 ) = 𝛼 + 𝛼
  • C c o s c o s c o s s i n s i n ( 𝛼 + 𝛽 ) = 𝛼 𝛽 βˆ’ 𝛼 𝛽 , s i n c o s s i n s i n c o s ( 𝛼 + 𝛽 ) = 𝛼 𝛽 + 𝛼 𝛽
  • D c o s c o s s i n c o s s i n ( 𝛼 + 𝛽 ) = 𝛼 𝛽 βˆ’ 𝛼 𝛽 , s i n s i n s i n s i n s i n ( 𝛼 + 𝛽 ) = 𝛼 𝛽 + 𝛼 𝛽

Q15:

Using the relation t a n s i n c o s πœƒ = πœƒ πœƒ , 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 ) β‰  πœ‹ 2 + πœ‹ 𝑛 .

  • A t a n t a n t a n t a n t a n ( 𝛼 + 𝛽 ) = 𝛼 + 𝛽 1 βˆ’ 𝛼 𝛽
  • B t a n t a n t a n t a n t a n ( 𝛼 + 𝛽 ) = 𝛼 + 𝛽 𝛼 βˆ’ 𝛽
  • C t a n t a n t a n t a n t a n ( 𝛼 + 𝛽 ) = 𝛼 βˆ’ 𝛽 1 βˆ’ 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 βˆ’ 𝛽 1 + 𝛼 𝛽

Q16:

Simplify t a n t a n t a n t a n 1 5 9 βˆ’ 1 1 4 1 + 1 5 9 1 1 4 ∘ ∘ ∘ ∘ .

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

Q17:

Simplify t a n t a n t a n t a n 1 5 6 βˆ’ 4 3 1 + 1 5 6 4 3 ∘ ∘ ∘ ∘ .

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

Q18:

Simplify t a n t a n t a n t a n 1 6 7 βˆ’ 1 0 2 1 + 1 6 7 1 0 2 ∘ ∘ ∘ ∘ .

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

Q19:

Simplify t a n t a n t a n t a n 2 2 βˆ’ 1 6 1 + 2 2 1 6 ∘ ∘ ∘ ∘ .

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