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How can the expression $4 ext{sin}(16x) ext{cos}(10x)$ be rewritten using the product-to-sum formulas?
How can the expression $4 ext{sin}(16x) ext{cos}(10x)$ be rewritten using the product-to-sum formulas?
$2 ext{sin}(26x) + 2 ext{sin}(6x)$
Using the product-to-sum formulas, what is the equivalent expression for $2 ext{cos}(5t) ext{sin}(3tx)$?
Using the product-to-sum formulas, what is the equivalent expression for $2 ext{cos}(5t) ext{sin}(3tx)$?
$ ext{sin}((5t + 3tx)) - ext{sin}((5t - 3tx))$
Rewrite the sum $sin(70°) + sin(30°)$ as a product of two functions.
Rewrite the sum $sin(70°) + sin(30°)$ as a product of two functions.
$2 ext{sin}(50°) ext{cos}(20°)$
Prove the identity for $sin(76°) - sin(14°)$ using the appropriate product-to-sum formula.
Prove the identity for $sin(76°) - sin(14°)$ using the appropriate product-to-sum formula.
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What is the result of converting $2 ext{cos}(56°) ext{sin}(48°)$ using product-to-sum formulas?
What is the result of converting $2 ext{cos}(56°) ext{sin}(48°)$ using product-to-sum formulas?
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How can you prove the identity $\cos(20^{\circ}) = \frac{\sin(70^{\circ}) \sin(50^{\circ}) \sin(30^{\circ}) \sin(10^{\circ})}{\cos(80^{\circ}) \cos(60^{\circ}) \cos(40^{\circ})}$?
How can you prove the identity $\cos(20^{\circ}) = \frac{\sin(70^{\circ}) \sin(50^{\circ}) \sin(30^{\circ}) \sin(10^{\circ})}{\cos(80^{\circ}) \cos(60^{\circ}) \cos(40^{\circ})}$?
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What is the result of simplifying $2\sin^2(y) \sin(3y) = \cos(y) - \cos(5y)$?
What is the result of simplifying $2\sin^2(y) \sin(3y) = \cos(y) - \cos(5y)$?
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In the expression $\sin(6\beta) + \sin(4\beta)$, how would you rewrite it using the sine addition formula?
In the expression $\sin(6\beta) + \sin(4\beta)$, how would you rewrite it using the sine addition formula?
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Describe how you could use the equation $2\tan(y) \cos(3y) = \sec(y)(\sin(4y) - \sin(2y))$ to solve for $y$.
Describe how you could use the equation $2\tan(y) \cos(3y) = \sec(y)(\sin(4y) - \sin(2y))$ to solve for $y$.
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What does the equation $\cos(6x) - \cos(10x) = \cot(2x) \cot(8x)$ signify in terms of angle properties?
What does the equation $\cos(6x) - \cos(10x) = \cot(2x) \cot(8x)$ signify in terms of angle properties?
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Study Notes
Exercise 8.3
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Product-to-Sum Formulas: Used to convert trigonometric products into sums or differences. Examples include:
- 4sin16xcos10x
- 10cos10ycos6y
- 2cos5tsin3tx
- 6cos5xsin10x
- sin(-u)sin5u
- 2sin(-100°)sin(-20°)
- cos23°sin17°
- 2cos56°sin48°
- 2sin75°sin15°
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Sum-to-Product Formulas: Used to convert trigonometric sums or differences into products. Examples include:
- sin70° + sin30°
- sin76° - sin14°
- cos58° + cos12°
- sin(-10°) + sin(-20°)
Proving Trigonometric Identities
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Identities:
- cos(α+β) / cos(α-β) = (1 - tanαtanβ) / (1 + tanαtanβ)
- 4cos4v sin3v = 2(sin7v - sinv)
- cos3x + cosx = 2cosx(cos2x)
- sin6β + sin4β = tan5β cotβ
- sin6β - sin4β = tan5β cotβ
- 6cos8usin2u -3sin10u / sin6u = sin6u / sin6u +3
- 2tanycos3y = secy(sin4y-sin2y)
Further Trigonometric Identities
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Additional Identities:
- cos6x + cos8x / sin 6x - sin4x = cotxcos7xsec5x
- cos2a - cos4a / sin2a + sin4a = tan a
- 2cos2ucosu +sin2usinu = 2cos³u
- 2sin2ysin3y = cos y - cos5y
- cos 10x + cos6x / cos6x - cos10x = cot2xcot8x
Proving Trigonometric Equations
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Equations:
- cos80° cos60° cos40° cos20° = 1/16
- sin70° sin50° sin30° sin10° = 1/16
- sin(π/9) - sin(2π/9) - sin(3π/9) - sin(4π/9) = 3 / 16
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Description
Test your knowledge on product-to-sum and sum-to-product formulas in trigonometry. This quiz also covers proving various trigonometric identities, ensuring a comprehensive understanding of the topic. Perfect for students looking to solidify their grasp on trigonometric principles.
