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Worksheet: Angle Sum and Difference Identities

Q1:

Given that s i n c o s c o s s i n s i n 6 0 3 0 βˆ’ 6 0 3 0 = πœƒ ∘ ∘ ∘ ∘ ∘ , find the value of πœƒ in degrees.

  • A 9 0 ∘
  • B 6 0 ∘
  • C 4 0 ∘
  • D 3 0 ∘
  • E 1 3 5 ∘

Q2:

Find s i n ( 𝐴 βˆ’ 𝐡 ) given s i n 𝐴 = βˆ’ 5 1 3 where 2 7 0 < 𝐴 < 3 6 0 ∘ ∘ and c o s 𝐡 = 4 5 where 0 < 𝐡 < 9 0 ∘ ∘ .

  • A 1 6 6 5
  • B βˆ’ 5 6 6 5
  • C βˆ’ 1 6 6 5
  • D βˆ’ 6 5 5 6
  • E 5 6 6 5

Q3:

Find the value of s i n ( 𝐡 βˆ’ 2 𝐴 ) given t a n 𝐴 = 4 3 where 𝐴 ∈ ο€» 0 , πœ‹ 2  and t a n 𝐡 = 2 4 7 where 𝐡 ∈ ο€Ό πœ‹ , 3 πœ‹ 2  .

  • A βˆ’ 4 4 1 2 5
  • B βˆ’ 3 3 6 6 2 5
  • C 4 4 1 2 5
  • D 3 3 6 6 2 5

Q4:

𝐴 𝐡 𝐢 is a triangle where c o s 𝐴 = 3 5 and s i n 𝐡 = 4 5 . Find s i n 𝐢 without using a calculator.

  • A βˆ’ 1
  • B 2 5 2 4
  • C0
  • D 2 4 2 5

Q5:

Find s i n ( 𝐴 βˆ’ 𝐡 ) given s i n 𝐴 = 4 5 where 9 0 < 𝐴 < 1 8 0 ∘ ∘ and c o s 𝐡 = βˆ’ 3 5 where 1 8 0 < 𝐡 < 2 7 0 ∘ ∘ .

  • A0
  • B βˆ’ 2 5 2 4
  • C 2 4 2 5
  • D βˆ’ 2 4 2 5

Q6:

Find c o s ( 𝐴 βˆ’ 𝐡 ) given c o s 𝐴 = 4 5 and c o s 𝐡 = 3 5 where 𝐴 and 𝐡 are acute angles.

  • A0
  • B 2 5 2 4
  • C βˆ’ 2 4 2 5
  • D 2 4 2 5

Q7:

Find s e c ( 𝐴 βˆ’ 𝐡 ) , given s i n 𝐴 = βˆ’ 4 5 where 1 8 0 < 𝐴 < 2 7 0 ∘ ∘ and c o s 𝐡 = 1 2 1 3 where 2 7 0 < 𝐡 < 3 6 0 ∘ ∘ .

  • A βˆ’ 5 6 6 5
  • B βˆ’ 1 6 6 5
  • C 1 6 6 5
  • D βˆ’ 6 5 1 6
  • E 5 6 6 5

Q8:

Find s i n ( 𝐴 + 𝐡 ) given c o s 𝐴 = 2 4 2 5 and t a n 𝐡 = 1 2 5 where 𝐴 and 𝐡 are two positive acute angles.

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

Q9:

Let us consider the two shown figures, each showing two points on the unit circle.

How is the figure on the right obtained from the figure on the left?

  • Aby rotating by an angle of βˆ’ 𝛼 about the origin
  • Bby rotating by an angle of 𝛽 about the origin
  • Cby rotating by an angle of 𝛼 about the origin
  • Dby rotating by an angle of βˆ’ 𝛽 about the origin
  • Eby rotating by an angle of 𝛽 βˆ’ 𝛼 about the origin

What can you say about the triangles 𝑂 𝑀 𝑁 and 𝑂 𝑀 β€² 𝑁 β€² ?

  • AThey are congruent.
  • BThey are isosceles.
  • CThey are equilateral.
  • DThey are similar.
  • EThey are different.

Find the coordinates of 𝑀 , 𝑁 , 𝑀 β€² , and 𝑁 β€² .

  • A 𝑀 ( 𝛽 , 𝛽 ) c o s s i n , 𝑁 ( 𝛼 , 𝛼 ) c o s s i n , 𝑀 β€² ( 0 , 1 ) , 𝑁 β€² ( ( 𝛼 βˆ’ 𝛽 ) , ( 𝛼 βˆ’ 𝛽 ) ) c o s s i n
  • B 𝑀 ( 𝛽 , 𝛽 ) s i n c o s , 𝑁 ( 𝛼 , 𝛼 ) s i n c o s , 𝑀 β€² ( 1 , 0 ) , 𝑁 β€² ( ( 𝛼 βˆ’ 𝛽 ) , ( 𝛼 βˆ’ 𝛽 ) ) c o s s i n
  • C 𝑀 ( 𝛽 , 𝛽 ) c o s s i n , 𝑁 ( 𝛼 , 𝛼 ) c o s s i n , 𝑀 β€² ( 1 , 0 ) , 𝑁 β€² ( ( 𝛼 βˆ’ 𝛽 ) , ( 𝛼 βˆ’ 𝛽 ) ) c o s s i n
  • D 𝑀 ( 𝛽 , 𝛽 ) c o s s i n , 𝑁 ( 𝛼 , 𝛼 ) c o s s i n , 𝑀 β€² ( 0 , 1 ) , 𝑁 β€² ( ( 𝛼 βˆ’ 𝛽 ) , ( 𝛼 βˆ’ 𝛽 ) ) s i n c o s
  • E 𝑀 ( 𝛽 , 𝛽 ) s i n c o s , 𝑁 ( 𝛼 , 𝛼 ) s i n c o s , 𝑀 β€² ( 0 , 1 ) , 𝑁 β€² ( ( 𝛼 βˆ’ 𝛽 ) , ( 𝛼 βˆ’ 𝛽 ) ) s i n c o s

Calculate the lengths of 𝑀 𝑁 and 𝑀 β€² 𝑁 β€² .

  • A 𝑀 𝑁 = √ 2 βˆ’ 2 𝛼 𝛽 βˆ’ 2 𝛼 𝛽 c o s c o s s i n s i n , 𝑀 β€² 𝑁 β€² = √ 2 βˆ’ 2 ( 𝛼 βˆ’ 𝛽 ) c o s
  • B 𝑀 𝑁 = 2 , 𝑀 β€² 𝑁 β€² = 2 βˆ’ 2 ( 𝛼 βˆ’ 𝛽 ) c o s
  • C 𝑀 𝑁 = √ 2 βˆ’ 2 𝛼 𝛽 βˆ’ 2 𝛼 𝛽 c o s c o s s i n s i n , 𝑀 β€² 𝑁 β€² = √ 2 + 2 ( 𝛼 βˆ’ 𝛽 ) c o s
  • D 𝑀 𝑁 = √ 2 , 𝑀 β€² 𝑁 β€² = √ 2 βˆ’ 2 ( 𝛼 βˆ’ 𝛽 ) c o s
  • E 𝑀 𝑁 = √ 2 , 𝑀 β€² 𝑁 β€² = √ 2 + 2 ( 𝛼 βˆ’ 𝛽 ) c o s

Use what you found in the previous questions to find an expression for c o s ( 𝛼 βˆ’ 𝛽 ) .

  • A c o s c o s c o s s i n s i n ( 𝛼 βˆ’ 𝛽 ) = 1 2 𝛼 𝛽 βˆ’ 1 2 𝛼 𝛽
  • B c o s c o s c o s s i n s i n ( 𝛼 βˆ’ 𝛽 ) = 𝛼 𝛽 βˆ’ 𝛼 𝛽
  • C c o s c o s c o s s i n s i n ( 𝛼 βˆ’ 𝛽 ) = βˆ’ 𝛼 𝛽 βˆ’ 𝛼 𝛽
  • D c o s c o s c o s s i n s i n ( 𝛼 βˆ’ 𝛽 ) = 𝛼 𝛽 + 𝛼 𝛽
  • E c o s s i n s i n c o s c o s ( 𝛼 βˆ’ 𝛽 ) = 𝛼 𝛽 βˆ’ 𝛼 𝛽

Q10:

Find c s c ( 𝐴 + 𝐡 ) given s i n 𝐴 = 4 5 where 9 0 < 𝐴 < 1 8 0 ∘ ∘ and c o s 𝐡 = βˆ’ 5 1 3 where 1 8 0 < 𝐡 < 2 7 0 ∘ ∘ .

  • A βˆ’ 5 6 6 5
  • B 1 6 6 5
  • C βˆ’ 1 6 6 5
  • D 6 5 1 6
  • E 5 6 6 5

Q11:

Evaluate t a n c o t c o t t a n 1 6 + 7 6 1 βˆ’ 7 6 1 6 ∘ ∘ ∘ ∘ .

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

Q12:

In triangle 𝐴 𝐡 𝐢 , 𝐴 and 𝐡 are acute angles, where s i n 𝐴 = 4 5 and c o s 𝐡 = 3 5 . Without using a calculator, find the value of c o s 𝐢 .

  • A 1 6 2 5
  • B βˆ’ 1 6 2 5
  • C 1 5
  • D 7 2 5
  • E βˆ’ 1 5

Q13:

Using the relation s i n s i n c o s c o s s i n ( 𝛼 βˆ’ 𝛽 ) = 𝛼 𝛽 βˆ’ 𝛼 𝛽 , find an expression for s i n ( 𝛼 + 𝛽 ) .

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

Q14:

Find s i n ( 𝐴 βˆ’ 𝐡 ) given s i n 𝐴 = βˆ’ 4 5 where 2 7 0 < 𝐴 < 3 6 0 ∘ ∘ and c o s 𝐡 = βˆ’ 4 5 where 1 8 0 < 𝐡 < 2 7 0 ∘ ∘ .

  • A βˆ’ 7 2 5
  • B βˆ’ 1
  • C 7 2 5
  • D1

Q15:

Find c s c ( 𝐴 βˆ’ 𝐡 ) given c o s 𝐴 = 7 2 5 and c o s 𝐡 = 3 5 where 𝐴 and 𝐡 are acute angles.

  • A 4 5
  • B 4 4 1 2 5
  • C βˆ’ 4 4 1 2 5
  • D 1 2 5 4 4
  • E βˆ’ 4 5

Q16:

Given that t a n πœƒ = 3 4 , where πœƒ is a positive acute angle, determine s i n ( πœƒ + 6 0 ) ∘ without using a calculator.

  • A 3 + 4 √ 3 5
  • B 3 βˆ’ 4 √ 3 1 0
  • C 3 5
  • D 3 + 4 √ 3 1 0

Q17:

Which of the following is equivalent to ?

  • A
  • B
  • C
  • D

Q18:

Given that t a n 𝐴 = 2 4 7 , where 0 < 𝐴 < 9 0 ∘ ∘ , and t a n 𝐡 = βˆ’ 8 1 5 , where 9 0 < 𝐡 < 1 8 0 ∘ ∘ , determine s i n ( 𝐴 βˆ’ 𝐡 ) .

  • A βˆ’ 3 0 4 4 2 5
  • B βˆ’ 4 2 5 4 1 6
  • C 3 0 4 4 2 5
  • D βˆ’ 4 1 6 4 2 5

Q19:

Evaluate t a n t a n t a n t a n ο€»  βˆ’ ο€»  1 + ο€»  ο€»  5 πœ‹ 6 2 πœ‹ 3 5 πœ‹ 6 2 πœ‹ 3 .

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

Q20:

Find s i n ( 𝐴 + 𝐡 ) given c o s 𝐴 = 4 5 and c o s 𝐡 = 3 5 where 𝐴 and 𝐡 are acute angles.

  • A 7 2 5
  • B βˆ’ 7 2 5
  • C βˆ’ 1
  • D1

Q21:

Given that s i n 𝐴 = 3 5 and c o s 𝐡 = 1 2 1 3 , where 𝐴 and 𝐡 are two positive acute angles, determine s i n ( 𝐴 + 𝐡 ) .

  • A 1 6 6 5
  • B 6 5 5 6
  • C βˆ’ 1 6 6 5
  • D 5 6 6 5

Q22:

Given that t a n 𝐴 = 4 3 and t a n 𝐡 = 7 2 4 , where 𝐴 and 𝐡 are two positive acute angles, determine s i n ( 𝐴 + 𝐡 ) .

  • A 3 5
  • B 1 2 5 1 1 7
  • C βˆ’ 3 5
  • D 1 1 7 1 2 5

Q23:

Find s i n ( 𝑋 βˆ’ π‘Œ ) given 2 5 𝑋 + 7 = 0 c o s where 9 0 < 𝑋 < 1 8 0 ∘ ∘ and c o s π‘Œ = 4 5 where 2 7 0 < π‘Œ < 3 6 0 ∘ ∘ .

  • A βˆ’ 3 5
  • B 5 3
  • C 1 1 7 1 2 5
  • D 3 5

Q24:

If s i n 2 𝐴 = 5 7 6 6 2 5 , where 1 8 0 < 𝐴 < 2 7 0 ∘ ∘ , and t a n 𝐡 = βˆ’ 4 3 , where 9 0 < 𝐡 < 1 8 0 ∘ ∘ , find s i n ( 𝐴 βˆ’ 𝐡 ) .

  • A βˆ’ 4 4 1 2 5
  • B 5 4
  • C 4 4 1 2 5
  • D 4 5

Q25:

Find given where and where .

  • A
  • B
  • C
  • D
  • E