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 sincoscossinsin60306030=𝜃, find the value of 𝜃 in degrees.

Q2:

Simplify tantantantan(1182𝑋)+(32+2𝑋)1(1182𝑋)(32+2𝑋).

  • A33
  • B3
  • C33
  • D3

Q3:

Evaluate tantantantan1+.

  • A3
  • B3
  • C33
  • D33

Q4:

Evaluate tancotcottan16+7617616.

  • A33
  • B3
  • C33
  • D3

Q5:

In triangle 𝐴𝐵𝐶, 𝐴 and 𝐵 are acute angles, where sin𝐴=45 and cos𝐵=35. Without using a calculator, find the value of cos𝐶.

  • A15
  • B1625
  • C725
  • D15
  • E1625

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

What can you say about the triangles 𝑂𝑀𝑁 and 𝑂𝑀𝑁?

  • 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((𝛼𝛽),(𝛼𝛽))sincos
  • B𝑀(𝛽,𝛽)cossin, 𝑁(𝛼,𝛼)cossin, 𝑀(1,0), 𝑁((𝛼𝛽),(𝛼𝛽))cossin
  • C𝑀(𝛽,𝛽)sincos, 𝑁(𝛼,𝛼)sincos, 𝑀(1,0), 𝑁((𝛼𝛽),(𝛼𝛽))cossin
  • D𝑀(𝛽,𝛽)cossin, 𝑁(𝛼,𝛼)cossin, 𝑀(0,1), 𝑁((𝛼𝛽),(𝛼𝛽))cossin
  • E𝑀(𝛽,𝛽)cossin, 𝑁(𝛼,𝛼)cossin, 𝑀(0,1), 𝑁((𝛼𝛽),(𝛼𝛽))sincos

Calculate the lengths of 𝑀𝑁and 𝑀𝑁.

  • A𝑀𝑁=2, 𝑀𝑁=22(𝛼𝛽)cos
  • B𝑀𝑁=22𝛼𝛽2𝛼𝛽coscossinsin, 𝑀𝑁=2+2(𝛼𝛽)cos
  • C𝑀𝑁=22𝛼𝛽2𝛼𝛽coscossinsin, 𝑀𝑁=22(𝛼𝛽)cos
  • D𝑀𝑁=2, 𝑀𝑁=2+2(𝛼𝛽)cos
  • E𝑀𝑁=2, 𝑀𝑁=22(𝛼𝛽)cos

Use what you found in the previous questions to find an expression for cos(𝛼𝛽).

  • Acossinsincoscos(𝛼𝛽)=𝛼𝛽𝛼𝛽
  • Bcoscoscossinsin(𝛼𝛽)=𝛼𝛽𝛼𝛽
  • Ccoscoscossinsin(𝛼𝛽)=𝛼𝛽+𝛼𝛽
  • Dcoscoscossinsin(𝛼𝛽)=12𝛼𝛽12𝛼𝛽
  • Ecoscoscossinsin(𝛼𝛽)=𝛼𝛽𝛼𝛽

Q7:

𝐴𝐵𝐶 is a triangle where cos𝐴=35 and sin𝐵=45. Find sin𝐶 without using a calculator.

  • A1
  • B0
  • C2524
  • D2425

Q8:

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

  • A0
  • B2524
  • C2425
  • D2425

Q9:

Using the relation sinsincoscossin(𝛼𝛽)=𝛼𝛽𝛼𝛽, find an expression for sin(𝛼+𝛽).

  • Asinsincoscossin(𝛼+𝛽)=𝛼𝛽+𝛼𝛽
  • Bsinsincoscossin(𝛼+𝛽)=𝛼𝛽+𝛼𝛽
  • Csinsincoscossin(𝛼+𝛽)=𝛼𝛽𝛼𝛽
  • Dsinsincoscossin(𝛼+𝛽)=𝛼𝛽𝛼𝛽

Q10:

Find csc(𝐴𝐵) given cos𝐴=725 and cos𝐵=35 where 𝐴 and 𝐵 are acute angles.

  • A44125
  • B12544
  • C45
  • D45
  • E44125

Q11:

Given that tan𝜃=34, where 𝜃 is a positive acute angle, determine sin(𝜃+60) without using a calculator.

  • A35
  • B34310
  • C3+4310
  • D3+435

Q12:

Which of the following is equivalent to 21+3?

  • Asin15
  • B415cos
  • Ctan15
  • D415sin

Q13:

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

  • A2425
  • B2425
  • C2524
  • D0

Q14:

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

  • A1665
  • B5665
  • C6556
  • D1665

Q15:

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

  • A44125
  • B44125
  • C54
  • D45

Q16:

Find cos(𝐴𝐵) given cos𝐴=45 and cos𝐵=35 where 𝐴 and 𝐵 are acute angles.

  • A2425
  • B2425
  • C0
  • D2524

Q17:

Using the relations coscoscossinsin(𝛼𝛽)=𝛼𝛽+𝛼𝛽 and coscos(𝛼(𝛽))=(𝛼+𝛽), find an expression for cos(𝛼+𝛽).

  • Acossinsincoscos(𝛼+𝛽)=𝛼𝛽𝛼𝛽
  • Bcoscoscossinsin(𝛼+𝛽)=𝛼𝛽+𝛼𝛽
  • Ccoscoscossinsin(𝛼+𝛽)=𝛼𝛽𝛼𝛽
  • Dcoscoscossinsin(𝛼+𝛽)=𝛼𝛽𝛼𝛽

Q18:

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

  • A171221
  • B22121
  • C21221
  • D171221
  • E21221

Q19:

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

  • A257
  • B725
  • C725
  • D1

Q20:

Using the relation coscoscossinsin(𝛼𝛽)=𝛼𝛽+𝛼𝛽, find an expression for cos(𝛼+𝛽).

  • Acossinsincoscos(𝛼+𝛽)=𝛼𝛽𝛼𝛽
  • Bcoscoscossinsin(𝛼+𝛽)=𝛼𝛽+𝛼𝛽
  • Ccoscoscossinsin(𝛼+𝛽)=𝛼𝛽𝛼𝛽
  • Dcoscoscossinsin(𝛼+𝛽)=𝛼𝛽𝛼𝛽

Q21:

In triangle 𝐴𝐵𝐶, 𝐴 and 𝐵 are acute angles, where sin𝐴=45, and cos𝐵=1213. Without using a calculator, find the value of cos𝐶.

  • A6516
  • B5665
  • C1665
  • D5665

Q22:

Given that sin𝜃=45, where 𝜃 is a positive acute angle, determine cos(150𝜃) without using a calculator.

  • A33+410
  • B33+410
  • C33+45
  • D35

Q23:

Evaluate tan(𝑥𝑦), given tantantantan𝑥𝑦1+𝑥𝑦=37.

Q24:

Find tan(𝐴𝐵) given cos𝐴=513 and cos𝐵=35 where 𝐴 and 𝐵 are two positive acute angles.

  • A5633
  • B89
  • C1663
  • D5633
  • E6316

Q25:

Find the value of tan(𝐴𝐵) given sin𝐴=1517 and sin𝐵=45 where 𝐴 and 𝐵 are two positive acute angles.

  • A7736
  • B7736
  • C1384
  • D1112
  • E8413

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