Worksheet: Angle Sum and Difference Identities

In this worksheet, we will practice deriving the angle sum and difference identities, graphically or using the unitary circle, and using them to find trigonometric values.

Q1:

Given that s i n c o s c o s s i n s i n 6 0 3 0 6 0 3 0 = 𝜃 , find the value of 𝜃 in degrees.

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

Q2:

Given that c o s 𝜃 = 5 7 , where 3 𝜋 2 𝜃 2 𝜋 , and c o s 𝜑 = 2 3 , where 0 𝜑 𝜋 2 , find the exact value of c o s ( 𝜑 𝜃 ) .

  • A 5 7 + 4 3 2 1
  • B 5 2 + 2 4 2 2 1
  • C 5 7 4 3 2 1
  • D 5 2 2 4 2 2 1
  • E 5 7 + 4 3 2 1

Q3:

Simplify t a n t a n t a n t a n ( 1 1 8 2 𝑋 ) + ( 3 2 + 2 𝑋 ) 1 ( 1 1 8 2 𝑋 ) ( 3 2 + 2 𝑋 ) .

  • A 3 3
  • B 3
  • C 3
  • D 3 3

Q4:

Evaluate t a n t a n t a n t a n 1 + 5 𝜋 6 2 𝜋 3 5 𝜋 6 2 𝜋 3 .

  • A 3
  • B 3 3
  • C 3
  • D 3 3

Q5:

Evaluate t a n c o t c o t t a n 1 6 + 7 6 1 7 6 1 6 .

  • A 3
  • B 3 3
  • C 3
  • D 3 3

Q6:

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

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

Q7:

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

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

Q8:

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 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 𝛽 about the origin
  • Eby rotating by an angle of 𝛽 𝛼 about the origin

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

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

Find the coordinates of 𝑀 , 𝑁 , 𝑀 , and 𝑁 .

  • A 𝑀 ( 𝛽 , 𝛽 ) c o s s i n , 𝑁 ( 𝛼 , 𝛼 ) c o s s i n , 𝑀 ( 0 , 1 ) , 𝑁 ( ( 𝛼 𝛽 ) , ( 𝛼 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 ( 1 , 0 ) , 𝑁 ( ( 𝛼 𝛽 ) , ( 𝛼 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 ( 1 , 0 ) , 𝑁 ( ( 𝛼 𝛽 ) , ( 𝛼 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 ( 0 , 1 ) , 𝑁 ( ( 𝛼 𝛽 ) , ( 𝛼 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 ( 0 , 1 ) , 𝑁 ( ( 𝛼 𝛽 ) , ( 𝛼 𝛽 ) ) s i n c o s

Calculate the lengths of 𝑀 𝑁 and 𝑀 𝑁 .

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

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

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

Q9:

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

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

Q10:

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

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

Q11:

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

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

Q12:

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

  • A 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 3 2 3 3 2 5

Q13:

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

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

Q14:

Using the relation s i n s i n c o s c o s s i n ( 𝛼 𝛽 ) = 𝛼 𝛽 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 ) = 𝛼 𝛽 + 𝛼 𝛽
  • B s i n s i n c o s c o s s i n ( 𝛼 + 𝛽 ) = 𝛼 𝛽 𝛼 𝛽
  • C s i n s i n c o s c o s s i n ( 𝛼 + 𝛽 ) = 𝛼 𝛽 𝛼 𝛽
  • D s i n s i n c o s c o s s i n ( 𝛼 + 𝛽 ) = 𝛼 𝛽 + 𝛼 𝛽

Q15:

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

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

Q16:

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

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

Q17:

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

  • A 3 + 4 3 5
  • B 3 4 3 1 0
  • C 3 5
  • D 3 + 4 3 1 0

Q18:

Which of the following is equivalent to 2 1 + 3 ?

  • A 4 1 5 c o s
  • B s i n 1 5
  • C t a n 1 5
  • D 4 1 5 s i n

Q19:

Given that t a n 𝐴 = 2 4 7 , where 0 < 𝐴 < 9 0 , and t a n 𝐵 = 8 1 5 , where 9 0 < 𝐵 < 1 8 0 , determine s i n ( 𝐴 𝐵 ) .

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

Q20:

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

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

Q21:

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

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

Q22:

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

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

Q23:

Find s i n ( 𝑋 𝑌 ) given 2 5 𝑋 + 7 = 0 c o s where 9 0 < 𝑋 < 1 8 0 and c o s 𝑌 = 4 5 where 2 7 0 < 𝑌 < 3 6 0 .

  • A 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 𝐵 = 4 3 , where 9 0 < 𝐵 < 1 8 0 , find s i n ( 𝐴 𝐵 ) .

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

Q25:

Find the value of s i n ( 𝐵 2 𝐴 ) given t a n 𝐴 = 4 3 where 𝐴 0 , 𝜋 2 and t a n 𝐵 = 2 4 7 where 𝐵 𝜋 , 3 𝜋 2 .

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

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