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If 3 cos x + 2 cos 3x = cos y, 3 sin x + 2 sin 3x = sin y, then the value of cos 2x is
- a)–1
- b)1/8
- c)–1/8
- d)7/8
Correct answer is option 'A'. Can you explain this answer?
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If 3 cos x + 2 cos 3x = cos y, 3 sin x + 2 sin 3x = sin y, then the va...
Explanation:
1. Given equations:
- 3 cos x + 2 cos 3x = cos y
- 3 sin x + 2 sin 3x = sin y
2. Use trigonometric identities:
- cos 3x = 4 cos^3 x - 3 cos x
- sin 3x = 3 sin x - 4 sin^3 x
3. Substitute these identities into the given equations:
- 3 cos x + 2(4 cos^3 x - 3 cos x) = cos y
- 3 sin x + 2(3 sin x - 4 sin^3 x) = sin y
4. Simplify the equations:
- 11 cos^3 x - 6 cos x = cos y
- 6 sin x - 8 sin^3 x = sin y
5. Use the trigonometric identity cos^2 x + sin^2 x = 1:
- cos^2 x = 1 - sin^2 x
6. Substitute cos^2 x into the equations:
- 11(1 - sin^2 x) - 6 cos x = cos y
- 6 sin x - 8 sin^3 x = sin y
7. Since cos y = cos 2x and sin y = sin 2x:
- 11 - 11 sin^2 x - 6 cos x = cos 2x
- 6 sin x - 8 sin^3 x = sin 2x
8. Use the identity sin 2x = 2 sin x cos x:
- 6 sin x - 8 sin^3 x = 2 sin x cos x
- 2 sin x(3 - 4 sin^2 x) = 2 sin x cos x
9. Since sin x ≠ 0:
- 3 - 4 sin^2 x = cos x
10. Use the identity cos 2x = 1 - 2 sin^2 x:
- cos 2x = 1 - 2(1 - cos x)
- cos 2x = 1 - 2 + 2 cos x
- cos 2x = -1
Therefore, the value of cos 2x is -1.
1. Given equations:
- 3 cos x + 2 cos 3x = cos y
- 3 sin x + 2 sin 3x = sin y
2. Use trigonometric identities:
- cos 3x = 4 cos^3 x - 3 cos x
- sin 3x = 3 sin x - 4 sin^3 x
3. Substitute these identities into the given equations:
- 3 cos x + 2(4 cos^3 x - 3 cos x) = cos y
- 3 sin x + 2(3 sin x - 4 sin^3 x) = sin y
4. Simplify the equations:
- 11 cos^3 x - 6 cos x = cos y
- 6 sin x - 8 sin^3 x = sin y
5. Use the trigonometric identity cos^2 x + sin^2 x = 1:
- cos^2 x = 1 - sin^2 x
6. Substitute cos^2 x into the equations:
- 11(1 - sin^2 x) - 6 cos x = cos y
- 6 sin x - 8 sin^3 x = sin y
7. Since cos y = cos 2x and sin y = sin 2x:
- 11 - 11 sin^2 x - 6 cos x = cos 2x
- 6 sin x - 8 sin^3 x = sin 2x
8. Use the identity sin 2x = 2 sin x cos x:
- 6 sin x - 8 sin^3 x = 2 sin x cos x
- 2 sin x(3 - 4 sin^2 x) = 2 sin x cos x
9. Since sin x ≠ 0:
- 3 - 4 sin^2 x = cos x
10. Use the identity cos 2x = 1 - 2 sin^2 x:
- cos 2x = 1 - 2(1 - cos x)
- cos 2x = 1 - 2 + 2 cos x
- cos 2x = -1
Therefore, the value of cos 2x is -1.
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If 3 cos x + 2 cos 3x = cos y, 3 sin x + 2 sin 3x = sin y, then the value of cos 2x isa)–1b)1/8c)–1/8d)7/8Correct answer is option 'A'. Can you explain this answer?
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