Trigonometrical Equations
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Questions and Answers

What is the general form of a trigonometric equation that can be reduced to a rational equation with respect to the arguments sin x, cos x, tan x, and cot x?

  • R(sin x, cos x, tan x, cot x) = 0
  • R(sin kx, cos nx, tan mx, cot lx) = 0 (correct)
  • R(sin x, cos x) = 0
  • R(sin nx, cos mx) = 0

    • What is the formula for cos x in terms of tan x?

  • cos x = 1 / (2 + tan^2 x)
  • cos x = 2 / (1 + tan^2 x)
  • cos x = 1 + tan x
  • cos x = 1 / (1 + tan^2 x) (correct)

    • What is the purpose of the DYGU formulae in trigonometry?

  • To reduce trigonometric equations to rational equations with respect to the arguments sin x, cos x, tan x, and cot x (correct)
  • To find the values of sine and cosine functions
  • To solve trigonometric equations with respect to x
  • To simplify trigonometric identities

    • What is the substitution used to reduce the trigonometric equation to a rational equation with respect to x?

    <p>t = tan x</p> Signup and view all the answers

    What is the general solution to the equation (cos x - sin x)(2 tan x + 1) + 2 = 0?

    <p>x = 2nπ ± π/3</p> Signup and view all the answers

    What is a trigonometric equation?

    <p>An equation involving one or more trigonometric ratios of an unknown angle</p> Signup and view all the answers

    What is a root of a trigonometric equation?

    <p>The value of the unknown angle that satisfies the equation</p> Signup and view all the answers

    What is the main difference between a trigonometric equation and a trigonometric identity?

    <p>A trigonometric equation is satisfied for some particular values of the unknown angle, while a trigonometric identity is true for every value of the unknown angle</p> Signup and view all the answers

    What is a particular solution of a trigonometric equation?

    <p>A specific value of the unknown angle that satisfies the equation</p> Signup and view all the answers

    What is the value of sin x cos x in terms of t, where sin x + cos x = t?

    <p>1/2(t2 - 1)</p> Signup and view all the answers

    Why do trigonometric equations generally have more than one solution?

    <p>Because trigonometric ratios are periodic in nature</p> Signup and view all the answers

    What is the solution to the equation tan x = 2 in the form x = nπ ±...?

    <p>nπ ± π/3</p> Signup and view all the answers

    What is the equation (t2 + 1)(1 - t2) = 0 equivalent to?

    <p>tan x = 2 and tan x = -2</p> Signup and view all the answers

    What is the form of the equation R(sin x + cos x, sin x cos x) = 0, where R is a rational function?

    <p>A polynomial equation in sin x and cos x</p> Signup and view all the answers

    What is the value of x that satisfies the equation tan x = 2, where x is in the form x = nπ ±...?

    <p>nπ ± π/3</p> Signup and view all the answers

    What can be concluded about the trigonometric equation |secθ + tanθ| = 2?

    <p>It has a particular solution θ = 60°.</p> Signup and view all the answers

    Which of the following statements is true about the trigonometric identity cos²x = 1 - sin²x?

    <p>It is true for every value of x in the domain of the function.</p> Signup and view all the answers

    What is the difference between a trigonometric equation and a trigonometric identity?

    <p>A trigonometric equation is satisfied for some particular values of the unknown angle, while a trigonometric identity is true for every value of the unknown angle.</p> Signup and view all the answers

    What is the name of the value of the unknown angle that satisfies the given trigonometric equation?

    <p>Root of the equation</p> Signup and view all the answers

    What is the characteristic of trigonometric ratios that results in trigonometric equations having more than one solution?

    <p>Periodicity</p> Signup and view all the answers

    If sin x + cos x = t, then what is the value of sin x cos x in terms of t?

    <p>(t2 - 1)/2</p> Signup and view all the answers

    What is the general form of the trigonometric equation that can be reduced to a rational equation with respect to the arguments sin x, cos x, tan x, and cot x?

    <p>R(sin x + cos x, sin x cos x) = 0</p> Signup and view all the answers

    If tan x = 2, then what is the value of x in the form x = nπ ±...?

    <p>x = nπ ± π/6</p> Signup and view all the answers

    What is the equivalent form of the equation (t2 + 1)(1 - t2) = 0, where t = sin x + cos x?

    <p>tan x = 2 or tan x = -2</p> Signup and view all the answers

    What is the value of x that satisfies the equation tan x = 2, where x is in the form x = nπ ±...?

    <p>x = nπ ± π/6</p> Signup and view all the answers

    If t = tan(x/2), then what is the value of sin(x) in terms of t?

    <p>2t/(1 + t^2)</p> Signup and view all the answers

    What is the value of cos(x) in terms of t, where t = tan(x) and x is in the first quadrant?

    <p>1/(t^2 + 1)</p> Signup and view all the answers

    If sin(x) + cos(x) = t, then what is the value of sin(x)cos(x) in terms of t?

    <p>(t^2 - 1)/2</p> Signup and view all the answers

    What is the general solution to the equation (cos(x) - sin(x))(2 tan(x) + 1) + 2 = 0?

    <p>x = nπ ± π/3</p> Signup and view all the answers

    If R(sin(x), cos(x), tan(x), cot(x)) = 0 is a trigonometric equation, what is the form of the equation after reducing it to a rational equation with respect to x?

    <p>R(t, 1/t, t^2, 1/t^2) = 0</p> Signup and view all the answers

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