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CBSE · Class 11 · Mathematics

Multiple Angles and Product-Sum Formulas (Ch 3: 3.4, part 2)

4 concepts · 32 practice questions with answers · Free to read

Double angle formulas

From daily life

A skateboarder doubles the turning: spin of 2x. Putting y = x in the sum formulas gives neat double-angle rules.

Simple definition

sin 2x = 2 sin x cos x; cos 2x = cos²x − sin²x = 2cos²x − 1 = 1 − 2 sin²x; tan 2x = 2 tan x/(1 − tan²x). Half-angle forms: cos²x = (1 + cos 2x)/2 and sin²x = (1 − cos 2x)/2.

Formula: sin 2x = 2 sin x cos x

Rule 1: Find 2x values — Get sin x, cos x first.

  • Easy: sin x = 3/5, x acute: sin 2x = 2(3/5)(4/5) = 24/25.
  • Medium: sin x = 3/5: cos 2x = 1 − 2(9/25) = 7/25.
  • Exam-type: tan x = 1/2: tan 2x = 1/(1 − 1/4) = 4/3.

Rule 2: Use half-angle forms — Turn squares into cos 2x.

  • Easy: cos²15° = (1 + cos 30°)/2 = (2 + √3)/4 ≈ 0.9330.
  • Medium: sin²15° = (1 − cos 30°)/2 ≈ 0.0670.
  • Exam-type: cos²22.5° = (1 + 1/√2)/2.
Remember

Three forms of cos 2x: choose the one that has only the quantity you know.

Common mistake

Writing sin 2x = 2 sin x.

Memory clue

Double sin: Two Sines Cos.

Practice questions

Easy
  1. 1Find sin 2x if sin x = 5/13, x acute.
  2. 2Find cos 2x if cos x = 4/5.
  3. 3Find tan 2x if tan x = 1/3.
Medium
  1. 4Simplify 2 sin 15° cos 15°.
  2. 5Simplify cos²22.5° − sin²22.5°.
  3. 6Prove: (1 − cos 2x)/(1 + cos 2x) = tan²x.
Challenging
  1. 7Prove: sin 2x/(1 + cos 2x) = tan x.
  2. 8If sin x + cos x = 1/2, find sin 2x.
Show answers (try them yourself first)
  1. 12(5/13)(12/13) = 120/169
  2. 22(16/25) − 1 = 7/25
  3. 3(2/3)/(1 − 1/9) = 3/4
  4. 4sin 30° = 1/2
  5. 5cos 45° = 1/√2
  6. 62 sin²x/(2 cos²x)
  7. 72 sin x cos x/(2 cos²x)
  8. 8Square: 1 + sin 2x = 1/4, so sin 2x = −3/4

Triple angle formulas

From daily life

Turning three times as far: write 3x = 2x + x and expand.

Simple definition

sin 3x = 3 sin x − 4 sin³x, cos 3x = 4 cos³x − 3 cos x, tan 3x = (3 tan x − tan³x)/(1 − 3 tan²x).

Formula: sin 3x = 3 sin x − 4 sin³x

Rule 1: Evaluate — Use a known x.

  • Easy: x = 30°: sin 90° = 3(1/2) − 4(1/8) = 1.
  • Medium: x = 10°: sin 30° = 3 sin 10° − 4 sin³10° gives 1/2.
  • Exam-type: x = 20°: cos 60° = 4cos³20° − 3 cos 20° = 1/2.

Rule 2: Prove — Expand sin(2x + x).

  • Easy: sin 3x = sin 2x cos x + cos 2x sin x.
  • Medium: = 2 sin x cos²x + (1 − 2 sin²x) sin x.
  • Exam-type: = 3 sin x − 4 sin³x.
Remember

The sine formula uses only sin, the cosine formula uses only cos.

Common mistake

Mixing signs: 3 sin x − 4 sin³x, not 4 sin x − 3 sin³x.

Memory clue

3 then 4: sin starts with 3, cos starts with 4.

Practice questions

Easy
  1. 1Find sin 3x if sin x = 1/2.
  2. 2Find cos 3x if cos x = 1/2.
  3. 3Find tan 3x if tan x = 1.
Medium
  1. 4Prove: sin x · sin(60° − x) · sin(60° + x) = (1/4) sin 3x.
  2. 5Find the value of cos 3x if cos x = 3/5.
  3. 6Using sin 3x formula check x = 90°.
Challenging
  1. 7If sin x = 0.6, find sin 3x.
  2. 8Solve 3 sin x − 4 sin³x = 0 for 0 < x < π.
Show answers (try them yourself first)
  1. 13/2 − 4/8 = 1
  2. 24/8 − 3/2 = −1
  3. 3(3 − 1)/(1 − 3) = −1
  4. 4(3 sin x − 4 sin³x)/4
  5. 54(27/125) − 9/5 = 108/125 − 225/125 = −117/125
  6. 63 − 4 = −1 = sin 270° ✓
  7. 71.8 − 4(0.216) = 0.936
  8. 8sin 3x = 0, so x = π/3, 2π/3

Sum and difference to product

From daily life

Two sound waves added together make a wave with a changing loudness (beats). This is the idea behind the sum-to-product formulas.

Simple definition

cos x + cos y = 2 cos((x + y)/2) cos((x − y)/2); cos x − cos y = −2 sin((x + y)/2) sin((x − y)/2); sin x + sin y = 2 sin((x + y)/2) cos((x − y)/2); sin x − sin y = 2 cos((x + y)/2) sin((x − y)/2).

Formula: sin x + sin y = 2 sin((x + y)/2) cos((x − y)/2)

Rule 1: Convert a sum to a product — Half of the sum, half of the difference.

  • Easy: sin 75° + sin 15° = 2 sin 45° cos 30°.
  • Medium: cos 70° + cos 10° = 2 cos 40° cos 30°.
  • Exam-type: sin 5x − sin 3x = 2 cos 4x sin x.

Rule 2: Evaluate — The product often gives a known value.

  • Easy: sin 75° + sin 15° = 2(1/√2)(√3/2) = √6/2 ≈ 1.2247.
  • Medium: cos 75° + cos 15° = 2 cos 45° cos 30° = √6/2.
  • Exam-type: cos 15° − cos 75° = −2 sin 45° sin(−30°) = √2/2.
Remember

Sum-to-product turns a hard sum into a product that is easier to factor or simplify.

Common mistake

Losing the negative sign in cos x − cos y.

Memory clue

Half the sum, half the difference.

Practice questions

Easy
  1. 1Write sin 7x + sin 3x as a product.
  2. 2Write cos 5x + cos x as a product.
  3. 3Write cos 3x − cos x as a product.
Medium
  1. 4Find the value of sin 105° + sin 15°.
  2. 5Simplify (sin 5x + sin 3x)/(cos 5x + cos 3x).
  3. 6Show that (cos x − cos y)/(sin x + sin y) = −tan((x − y)/2).
Challenging
  1. 7Prove: sin 3x + sin x = 2 sin 2x cos x.
  2. 8Simplify (sin x + sin 3x)/(cos x + cos 3x).
Show answers (try them yourself first)
  1. 12 sin 5x cos 2x
  2. 22 cos 3x cos 2x
  3. 3−2 sin 2x sin x
  4. 42 sin 60° cos 45° = √6/2 ≈ 1.2247
  5. 52 sin 4x cos x / (2 cos 4x cos x) = tan 4x
  6. 6Numerator = −2 sin((x+y)/2) sin((x−y)/2); denominator = 2 sin((x+y)/2) cos((x−y)/2); divide.
  7. 7Half sum = 2x, half difference = x
  8. 8tan 2x

Product to sum

From daily life

Going the other way: a product of waves is a sum of two simpler waves.

Simple definition

2 cos x cos y = cos(x + y) + cos(x − y); −2 sin x sin y = cos(x + y) − cos(x − y); 2 sin x cos y = sin(x + y) + sin(x − y); 2 cos x sin y = sin(x + y) − sin(x − y).

Formula: 2 sin x cos y = sin(x + y) + sin(x − y)

Rule 1: Expand a product — Write 2·(product) first and then divide by 2.

  • Easy: sin 50° cos 10° = (1/2)(sin 60° + sin 40°).
  • Medium: cos 4x cos 2x = (1/2)(cos 6x + cos 2x).
  • Exam-type: sin 3x sin x = (1/2)(cos 2x − cos 4x).

Rule 2: Evaluate — Use values of standard angles.

  • Easy: sin 75° cos 15° = (1/2)(sin 90° + sin 60°) = (2 + √3)/4 ≈ 0.9330.
  • Medium: cos 75° cos 15° = (1/2)(cos 90° + cos 60°) = 1/4.
  • Exam-type: sin 75° sin 15° = (1/2)(cos 60° − cos 90°) = 1/4.
Remember

Half the sum of two simple waves.

Common mistake

Forgetting the factor 1/2 when you remove the 2.

Memory clue

Product → Plus.

Practice questions

Easy
  1. 1Write 2 sin 4x cos 2x as a sum.
  2. 2Write cos 5x cos 3x as a sum.
  3. 3Find sin 15° cos 15°.
Medium
  1. 4Find cos 20° cos 40° cos 80°.
  2. 5Find sin 20° sin 40° sin 80°.
  3. 6Simplify 2 cos 3x sin x.
Challenging
  1. 7Prove: cos(x + y) + cos(x − y) = 2 cos x cos y.
  2. 8Evaluate sin 105° cos 15°.
Show answers (try them yourself first)
  1. 1sin 6x + sin 2x
  2. 2(1/2)(cos 8x + cos 2x)
  3. 3(1/2) sin 30° = 1/4
  4. 41/8
  5. 5√3/8
  6. 6sin 4x − sin 2x
  7. 7Expand both and add
  8. 8(1/2)(sin 120° + sin 90°) = (2 + √3)/4 ≈ 0.9330