Find the arc length from (-3,4) clockwise to (4,3) along the circle
x^(2) + y^(2) = 25. Show that the result is one-fourth the circumference of the circle.
2007-02-01 12:37:29 · 3 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
The arc of a circle subtending an angle theta at center has its length = theta(in radian)* radius
In your given case, the angle subtended at centre is in two parts;
One part lies in the 4th quadrant and makes angle theta1 with y-axis and whose cosine = cos theta1 = 4/5 = 0.8
giving theta1 = 36.87 deg
The second part lies in the 1st quadrant and makes angle theta2 with y-axis, whose sine = sin theta2 = 4/5 = 0.8,
giving theta2 = 53.13 deg
Total angle subtended at centre = theta1 + theta2 = 90 deg
Length of arc therefore = pi/2 * r = 15.7
Since arc subtends angle of 90 deg or 1/4 of 360 deg, length of arc = 1/4 of circumference of circle
2007-02-08 22:13:35 · answer #1 · answered by Paleologus 3 · 0⤊ 0⤋
The radius to (-3,4) is perpendicular to the radius to (4,3) (the slopes are negative reciprocals). Therefor the minor arc length between the points is (Ï/2)*5, which is 1/4 of 10Ï.
2007-02-01 20:50:37 · answer #2 · answered by Helmut 7 · 1⤊ 0⤋
RTFB
2007-02-01 20:41:20 · answer #3 · answered by Steelhead 5 · 0⤊ 3⤋