Trigonometric Identities Practice

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Questions and Answers

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 $2 ext{cos}(5t) ext{sin}(3tx)$?

$ ext{sin}((5t + 3tx)) - ext{sin}((5t - 3tx))$

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.

<p>$2 ext{cos}(45°) ext{sin}(31°)$</p> Signup and view all the answers

What is the result of converting $2 ext{cos}(56°) ext{sin}(48°)$ using product-to-sum formulas?

<p>$ ext{sin}(104°) - ext{sin}(8°)$</p> Signup and view all the answers

How can you prove the identity $\cos(20^{\circ}) = \frac{\sin(70^{\circ}) \sin(50^{\circ}) \sin(30^{\circ}) \sin(10^{\circ})}{\cos(80^{\circ}) \cos(60^{\circ}) \cos(40^{\circ})}$?

<p>You can use trigonometric identities and angle transformations to manipulate both sides until they align.</p> Signup and view all the answers

What is the result of simplifying $2\sin^2(y) \sin(3y) = \cos(y) - \cos(5y)$?

<p>The simplified form is an identity that demonstrates the relationship between sine and cosine functions.</p> Signup and view all the answers

In the expression $\sin(6\beta) + \sin(4\beta)$, how would you rewrite it using the sine addition formula?

<p>You would apply the identity $\sin(A) + \sin(B) = 2 \sin(\frac{A+B}{2}) \cos(\frac{A-B}{2})$.</p> Signup and view all the answers

Describe how you could use the equation $2\tan(y) \cos(3y) = \sec(y)(\sin(4y) - \sin(2y))$ to solve for $y$.

<p>You would isolate one side to express $y$ in terms of known functions, often requiring iterative methods or numerical solutions.</p> Signup and view all the answers

What does the equation $\cos(6x) - \cos(10x) = \cot(2x) \cot(8x)$ signify in terms of angle properties?

<p>It indicates a relationship that can be explored through angle subtraction identities and can demonstrate periodicity in functions.</p> Signup and view all the answers

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Flashcards

Product-to-Sum Identity (sin*sin)

Transforms the product of two sines into a difference of cosines.

Product-to-Sum Formula

A trigonometric identity that transforms a product of two trigonometric functions into a sum or difference of two trigonometric functions.

Product-to-Sum Identity (cos*cos)

Transforms the product of two cosines into a sum of cosines.

Product-to-Sum Identity (sin*cos)

Transforms the product of a sine and a cosine into a sum of sines.

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Sum-to-Product Formula

The transformation of a sum or difference of trigonometric functions into a product of two trigonometric functions.

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cos(a + b) identity

This trigonometric identity states that the cosine of the sum of two angles (a + b) can be expressed in terms of the tangent of the individual angles.

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cos6x + cos8x

This identity expresses the sum of cosines of two angles (6x and 8x) in terms of cotangent, cosine, and secant functions.

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sin 3v = 2(sin 7v - sin v)

This identity demonstrates the relationship between the sine of thrice an angle (3v) and the sine of 7 times the angle minus the sine of the angle itself (7v - v).

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cos 20⁰ = (sin 70⁰ sin 50⁰ sin 30⁰ sin 10⁰) / (cos 80⁰ cos 60⁰ cos 40⁰)

This identity proves that the cosine of 20 degrees can be expressed as a product of various trigonometric functions.

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prove that 2cos2u cosu + sin2u sin u = 2cos³u

This identity establishes an equivalence between the left-hand side and right-hand side expressions involving trigonometric functions of various arguments.

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Study Notes

Exercise 8.3

  • 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°
  • 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

  • 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

  • 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

  • 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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