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Trigonometry

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I am trying to help my child out with a problem but yeah I am finding it veyr hard.

Gabe is spending two weeks on an archaeological dig. He finds a fragment of a circular plate that his leader thinks may be valuable. The arc length of the fragment is about 1/6 the circumference of the original complete plate and measures 1.65 inches. A similar plate found earlier has a diameter of 3.14 inches.

Write an an expression for the radius, r, of the earlier plate.
r=2π
r=π/4
r=π/3
r=π/2

What is the measure, in radians, of a central angle, θ, that intercepts an arc that is 1/6 the length of the circumference of a circle?
θ=2π/6
θ=2π/5
θ=2π/4
θ=2π/3

Write an expression for the arc length, S, intercepted by this central angle

Could Gabe's plate be a matching plate? Explain.

  • Trigonometry -

    Write an an expression for the radius, r, of the earlier plate.

    The earlier plate has a diameter of 3.14 inches, approximately equal to π inches.
    The radius equals half the diameter.

    What is the measure, in radians, of a central angle, θ, that intercepts an arc that is 1/6 the length of the circumference of a circle?
    The central angle of a complete circle is 360°, or 2π radians.
    So the central angle of an arc equal to 1/6 of the circumference is therefore 1/6th of the complete circle. Therefore the central angle is (1/6)*2π.

    Write an expression for the arc length, S, intercepted by this central angle
    The arc length of radius r and central angle θ is rθ. Since r and θ are known, solve for arc length.

    Could Gabe's plate be a matching plate? Explain.
    It is most likely a matching plate, because the 1/6th of the arc of the previous plate has an arc length of 3.14*π/6=1.643, well within the accuracy expected of the later plate.

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