Lesson Worksheet: Angle Sum and Difference Identities Mathematics

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 sincoscossinsin6030βˆ’6030=πœƒβˆ˜βˆ˜βˆ˜βˆ˜βˆ˜, find the value of πœƒ in degrees.

Q2:

Simplify tantantantan(118βˆ’2𝑋)+(32+2𝑋)1βˆ’(118βˆ’2𝑋)(32+2𝑋)∘∘∘∘.

  • Aβˆ’βˆš33
  • Bβˆ’βˆš3
  • C√33
  • D√3

Q3:

Evaluate tantantantanο€»ο‡βˆ’ο€»ο‡1+ο€»ο‡ο€»ο‡οŠ«οŽ„οŠ¬οŠ¨οŽ„οŠ©οŠ«οŽ„οŠ¬οŠ¨οŽ„οŠ©.

  • Aβˆ’βˆš3
  • B√3
  • C√33
  • Dβˆ’βˆš33

Q4:

Evaluate tancotcottan16+761βˆ’7616∘∘∘∘.

  • Aβˆ’βˆš33
  • Bβˆ’βˆš3
  • C√33
  • D√3

Q5:

In triangle 𝐴𝐡𝐢, 𝐴 and 𝐡 are acute angles, where sin𝐴=45 and cos𝐡=35. Without using a calculator, find the value of cos𝐢.

  • Aβˆ’15
  • B1625
  • C725
  • D15
  • Eβˆ’1625

Q6:

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 𝛽 about the origin
  • Bby rotating by an angle of βˆ’π›Ό about the origin
  • Cby rotating by an angle of βˆ’π›½ about the origin
  • Dby rotating by an angle of π›½βˆ’π›Ό about the origin
  • Eby rotating by an angle of 𝛼 about the origin

What can you say about the triangles 𝑂𝑀𝑁 and 𝑂𝑀′𝑁′?

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

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

  • A𝑀(𝛽,𝛽)sincos, 𝑁(𝛼,𝛼)sincos, 𝑀′(0,1), 𝑁′((π›Όβˆ’π›½),(π›Όβˆ’π›½))sincos
  • B𝑀(𝛽,𝛽)cossin, 𝑁(𝛼,𝛼)cossin, 𝑀′(1,0), 𝑁′((π›Όβˆ’π›½),(π›Όβˆ’π›½))cossin
  • C𝑀(𝛽,𝛽)sincos, 𝑁(𝛼,𝛼)sincos, 𝑀′(1,0), 𝑁′((π›Όβˆ’π›½),(π›Όβˆ’π›½))cossin
  • D𝑀(𝛽,𝛽)cossin, 𝑁(𝛼,𝛼)cossin, 𝑀′(0,1), 𝑁′((π›Όβˆ’π›½),(π›Όβˆ’π›½))cossin
  • E𝑀(𝛽,𝛽)cossin, 𝑁(𝛼,𝛼)cossin, 𝑀′(0,1), 𝑁′((π›Όβˆ’π›½),(π›Όβˆ’π›½))sincos

Calculate the lengths of 𝑀𝑁and 𝑀′𝑁′.

  • A𝑀𝑁=2, 𝑀′𝑁′=2βˆ’2(π›Όβˆ’π›½)cos
  • B𝑀𝑁=√2βˆ’2π›Όπ›½βˆ’2𝛼𝛽coscossinsin, 𝑀′𝑁′=√2+2(π›Όβˆ’π›½)cos
  • C𝑀𝑁=√2βˆ’2π›Όπ›½βˆ’2𝛼𝛽coscossinsin, 𝑀′𝑁′=√2βˆ’2(π›Όβˆ’π›½)cos
  • D𝑀𝑁=√2, 𝑀′𝑁′=√2+2(π›Όβˆ’π›½)cos
  • E𝑀𝑁=√2, 𝑀′𝑁′=√2βˆ’2(π›Όβˆ’π›½)cos

Use what you found in the previous questions to find an expression for cos(π›Όβˆ’π›½).

  • Acossinsincoscos(π›Όβˆ’π›½)=π›Όπ›½βˆ’π›Όπ›½
  • Bcoscoscossinsin(π›Όβˆ’π›½)=βˆ’π›Όπ›½βˆ’π›Όπ›½
  • Ccoscoscossinsin(π›Όβˆ’π›½)=𝛼𝛽+𝛼𝛽
  • Dcoscoscossinsin(π›Όβˆ’π›½)=12π›Όπ›½βˆ’12𝛼𝛽
  • Ecoscoscossinsin(π›Όβˆ’π›½)=π›Όπ›½βˆ’π›Όπ›½

Q7:

𝐴𝐡𝐢 is a triangle where cos𝐴=35 and sin𝐡=45. Find sin𝐢 without using a calculator.

  • Aβˆ’1
  • B0
  • C2524
  • D2425

Q8:

Find sin(𝐴+𝐡) given cos𝐴=45 and tan𝐡=34 where 𝐴 and 𝐡 are two positive acute angles.

  • A0
  • B2524
  • Cβˆ’2425
  • D2425

Q9:

Using the relation sinsincoscossin(π›Όβˆ’π›½)=π›Όπ›½βˆ’π›Όπ›½, find an expression for sin(𝛼+𝛽).

  • Asinsincoscossin(𝛼+𝛽)=𝛼𝛽+𝛼𝛽
  • Bsinsincoscossin(𝛼+𝛽)=βˆ’π›Όπ›½+𝛼𝛽
  • Csinsincoscossin(𝛼+𝛽)=π›Όπ›½βˆ’π›Όπ›½
  • Dsinsincoscossin(𝛼+𝛽)=βˆ’π›Όπ›½βˆ’π›Όπ›½

Q10:

Find csc(π΄βˆ’π΅) given cos𝐴=725 and cos𝐡=35 where 𝐴 and 𝐡 are acute angles.

  • Aβˆ’44125
  • B12544
  • Cβˆ’45
  • D45
  • E44125

Q11:

Given that tanπœƒ=34, where πœƒ is a positive acute angle, determine sin(πœƒ+60)∘ without using a calculator.

  • A35
  • B3βˆ’4√310
  • C3+4√310
  • D3+4√35

Q12:

Which of the following is equivalent to √2ο€»βˆ’1+√3?

  • Asin15∘
  • B415cos∘
  • Ctan15∘
  • D415sin∘

Q13:

Find csc(𝐴+𝐡) given cos𝐴=35 and cos𝐡=35 where 𝐴 and 𝐡 are acute angles.

  • A2425
  • Bβˆ’2425
  • C2524
  • D0

Q14:

Given that sin𝐴=35 and cos𝐡=1213, where 𝐴 and 𝐡 are two positive acute angles, determine sin(𝐴+𝐡).

  • A1665
  • B5665
  • C6556
  • Dβˆ’1665

Q15:

Given that tan𝐴=724 and tan𝐡=34, where 𝐴 and 𝐡 are two positive acute angles, determine sin(𝐴+𝐡).

  • A44125
  • Bβˆ’44125
  • C54
  • D45

Q16:

Find cos(π΄βˆ’π΅) given cos𝐴=45 and cos𝐡=35 where 𝐴 and 𝐡 are acute angles.

  • Aβˆ’2425
  • B2425
  • C0
  • D2524

Q17:

Using the relations coscoscossinsin(π›Όβˆ’π›½)=𝛼𝛽+𝛼𝛽 and coscos(π›Όβˆ’(βˆ’π›½))=(𝛼+𝛽), find an expression for cos(𝛼+𝛽).

  • Acossinsincoscos(𝛼+𝛽)=π›Όπ›½βˆ’π›Όπ›½
  • Bcoscoscossinsin(𝛼+𝛽)=𝛼𝛽+𝛼𝛽
  • Ccoscoscossinsin(𝛼+𝛽)=π›Όπ›½βˆ’π›Όπ›½
  • Dcoscoscossinsin(𝛼+𝛽)=βˆ’π›Όπ›½βˆ’π›Όπ›½

Q18:

Find cos(𝐴+𝐡) given cos𝐴=1517 and cos𝐡=513 where 𝐴 and 𝐡 are acute angles.

  • A171221
  • Bβˆ’22121
  • C21221
  • Dβˆ’171221
  • Eβˆ’21221

Q19:

Given that sin𝐴=35 and cos𝐡=45, where 𝐴 and 𝐡 are two positive acute angles, determine cos(𝐴+𝐡).

  • A257
  • Bβˆ’725
  • C725
  • D1

Q20:

Using the relation coscoscossinsin(π›Όβˆ’π›½)=𝛼𝛽+𝛼𝛽, find an expression for cos(𝛼+𝛽).

  • Acossinsincoscos(𝛼+𝛽)=π›Όπ›½βˆ’π›Όπ›½
  • Bcoscoscossinsin(𝛼+𝛽)=𝛼𝛽+𝛼𝛽
  • Ccoscoscossinsin(𝛼+𝛽)=π›Όπ›½βˆ’π›Όπ›½
  • Dcoscoscossinsin(𝛼+𝛽)=βˆ’π›Όπ›½βˆ’π›Όπ›½

Q21:

In triangle 𝐴𝐡𝐢, 𝐴 and 𝐡 are acute angles, where sin𝐴=45, and cos𝐡=1213. Without using a calculator, find the value of cos𝐢.

  • Aβˆ’6516
  • B5665
  • Cβˆ’1665
  • Dβˆ’5665

Q22:

Given that sinπœƒ=45, where πœƒ is a positive acute angle, determine cos(150βˆ’πœƒ)∘ without using a calculator.

  • Aβˆ’3√3+410
  • Bβˆ’3√3+410
  • Cβˆ’3√3+45
  • D35

Q23:

Evaluate tan(π‘₯βˆ’π‘¦), given tantantantanπ‘₯βˆ’π‘¦1+π‘₯𝑦=37.

Q24:

Find tan(π΄βˆ’π΅) given cos𝐴=513 and cos𝐡=35 where 𝐴 and 𝐡 are two positive acute angles.

  • A5633
  • Bβˆ’89
  • C1663
  • Dβˆ’5633
  • E6316

Q25:

Find the value of tan(π΄βˆ’π΅) given sin𝐴=1517 and sin𝐡=45 where 𝐴 and 𝐡 are two positive acute angles.

  • Aβˆ’7736
  • B7736
  • C1384
  • Dβˆ’1112
  • E8413

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