Educational

Find the angle in radians through which a pendulum swings if its length is \(75\) cm and the tip describes an arc of length \(10\) cm. Solution: We know: \[ s=r\theta \] Given: \[ s=10 \text{ cm} \] \[ r=75 \text{ cm} \] Using, \[ \theta=\frac{s}{r} \] \[ \theta=\frac{10}{75} \] \[ \theta=\frac{2}{15} \] Therefore, the […]

A wheel makes \(360\) revolutions per minute. Through how many radians does it turn in \(1\) second? Solution: Number of revolutions per minute: \[ 360 \] Number of revolutions per second: \[ \frac{360}{60}=6 \] One complete revolution equals: \[ 2\pi \text{ radians} \] Therefore, angle turned in \(1\) second: \[ 6\times2\pi \] \[ 12\pi \text{

Find the length which at a distance of \(5280\) m will subtend an angle of \(1’\) at the eye. Solution: We know: \[ s=r\theta \] Given: \[ r=5280 \text{ m} \] \[ \theta=1′ \] Since, \[ 1^\circ=60′ \] \[ 1’=\frac{1}{60}^\circ \] Convert into radians: \[ \theta=\frac{1}{60}\times\frac{\pi}{180} = \frac{\pi}{10800} \] Now, \[ s=5280\times\frac{\pi}{10800} \] \[ s=\frac{22\pi}{45}

A railroad curve is to be laid out on a circle. What radius should be used if the track is to change direction by \(25^\circ\) in a distance of \(40\) meters? Solution: We know: \[ \text{Arc Length} = r\theta \] Given: \[ s=40 \text{ m} \] \[ \theta=25^\circ \] Convert angle into radians: \[ 25^\circ

The number of sides of two regular polygons are as \(5:4\) and the difference between their angles is \(9^\circ\). Find the number of sides of the polygons. Solution: Let the number of sides of the polygons be: \[ 5x \text{ and } 4x \] Interior angle of a regular polygon: \[ \frac{(n-2)\times180^\circ}{n} \] First polygon

The angles of a triangle are in A.P. such that the greatest angle is 5 times the least. Find the angles in radians. Solution: Let the angles in A.P. be: \[ a-d,\ a,\ a+d \] Given, \[ a+d=5(a-d) \] \[ a+d=5a-5d \] \[ 6d=4a \] \[ 3d=2a \] Sum of angles of a triangle: \[

The angle in one regular polygon is to that in another as \(3:2\) and the number of sides in the first is twice that in the second. Determine the number of sides of the two polygons. Solution: Let the number of sides of the second polygon be \(n\). Then the number of sides of the

The angles of a triangle are in A.P. and the number of degrees in the least angle is to the number of degrees in the mean angle as \(1:2\). Find the angles in radians. Solution: Let the three angles in A.P. be: \[ a-d,\ a,\ a+d \] Given, \[ \frac{a-d}{a}=\frac{1}{2} \] \[ 2a-2d=a \] \[

The angles of a quadrilateral are in A.P. and the greatest angle is \(120^\circ\). Express the angles in radians. Solution: Let the four angles in A.P. be: \[ a-3d,\ a-d,\ a+d,\ a+3d \] Given greatest angle: \[ a+3d=120^\circ \] Sum of angles of a quadrilateral: \[ 360^\circ \] So, \[ (a-3d)+(a-d)+(a+d)+(a+3d)=360^\circ \] \[ 4a=360^\circ \]

Find the magnitude, in radians and degrees, of the interior angle of a regular duodecagon. Solution: A regular duodecagon has \(12\) sides. Interior angle of a regular polygon: \[ \frac{(n-2)\times180^\circ}{n} \] Substituting \(n=12\): \[ \frac{(12-2)\times180^\circ}{12} \] \[ \frac{10\times180^\circ}{12} \] \[ 150^\circ \] Now convert into radians: \[ 150^\circ \times \frac{\pi}{180^\circ} \] \[ \frac{5\pi}{6} \] Therefore,