Educational

Find the magnitude, in radians and degrees, of the interior angle of a regular heptagon. Solution: A regular heptagon has \(7\) sides. Interior angle of a regular polygon: \[ \frac{(n-2)\times180^\circ}{n} \] Substituting \(n=7\): \[ \frac{(7-2)\times180^\circ}{7} \] \[ \frac{900^\circ}{7} \] \[ 128\frac{4}{7}^\circ \] Now convert into radians: \[ \frac{900^\circ}{7}\times\frac{\pi}{180^\circ} \] \[ \frac{5\pi}{7} \] Therefore, the interior […]

Find the magnitude, in radians and degrees, of the interior angle of a regular octagon. Solution: A regular octagon has \(8\) sides. Interior angle of a regular polygon: \[ \frac{(n-2)\times180^\circ}{n} \] Substituting \(n=8\): \[ \frac{(8-2)\times180^\circ}{8} \] \[ \frac{6\times180^\circ}{8} \] \[ 135^\circ \] Now convert into radians: \[ 135^\circ \times \frac{\pi}{180^\circ} \] \[ \frac{3\pi}{4} \] Therefore,

Find the magnitude, in radians and degrees, of the interior angle of a regular pentagon. Solution: A regular pentagon has \(5\) sides. Interior angle of a regular polygon: \[ \frac{(n-2)\times180^\circ}{n} \] Substituting \(n=5\): \[ \frac{(5-2)\times180^\circ}{5} \] \[ \frac{3\times180^\circ}{5} \] \[ 108^\circ \] Now convert into radians: \[ 108^\circ \times \frac{\pi}{180^\circ} \] \[ \frac{3\pi}{5} \] Therefore,

One angle of a triangle is \( \frac{2x}{3} \) grade and another is \( \frac{3x}{2}^\circ \) while the third is \( \frac{\pi x}{75} \) radians. Express all the angles in degrees. Solution: We know: \[ 1 \text{ grade} = \frac{9^\circ}{10} \] and \[ 1 \text{ radian} = \frac{180^\circ}{\pi} \] First angle: \[ \frac{2x}{3} \text{ grade}

The difference between the two acute angles of a right-angled triangle is \( \frac{2\pi}{5} \) radians. Express the angles in degrees. Solution: In a right-angled triangle, the sum of the two acute angles is \(90^\circ\). Let the acute angles be \(A\) and \(B\). \[ A + B = 90^\circ \] Given, \[ A – B

Find the Radian Measure Corresponding to the Following Degree Measure : \( -47^\circ 30′ \) First convert minutes into degrees: \[ 30′ = \frac{30}{60}^\circ = \frac{1}{2}^\circ \] Therefore, \[ -47^\circ 30′ = -47.5^\circ = -\frac{95}{2}^\circ \] To convert a degree measure into radians, we use the formula: \[ \text{Radian} = \text{Degree} \times \frac{\pi}{180^\circ} \] Conversion

Find the Radian Measure Corresponding to the Following Degree Measure : \( 125^\circ 30′ \) First convert minutes into degrees: \[ 30′ = \frac{30}{60}^\circ = \frac{1}{2}^\circ \] Therefore, \[ 125^\circ 30′ = 125.5^\circ = \frac{251}{2}^\circ \] To convert a degree measure into radians, we use the formula: \[ \text{Radian} = \text{Degree} \times \frac{\pi}{180^\circ} \] Conversion

Find the Radian Measure Corresponding to the Following Degree Measure : \( 7^\circ 30′ \) First convert minutes into degrees: \[ 30′ = \frac{30}{60}^\circ = \frac{1}{2}^\circ \] Therefore, \[ 7^\circ 30′ = 7.5^\circ = \frac{15}{2}^\circ \] To convert a degree measure into radians, we use the formula: \[ \text{Radian} = \text{Degree} \times \frac{\pi}{180^\circ} \] Conversion

Find the Radian Measure Corresponding to the Following Degree Measure : \( -300^\circ \) To convert a degree measure into radians, we use the formula: \[ \text{Radian} = \text{Degree} \times \frac{\pi}{180^\circ} \] Given: \[ -300^\circ \] Conversion into Radians \[ -300^\circ \times \frac{\pi}{180^\circ} \] \[ = \frac{-300\pi}{180} \] \[ = -\frac{5\pi}{3} \] Answer \[ -\frac{5\pi}{3}

Find the Radian Measure Corresponding to the Following Degree Measure : \( 135^\circ \) To convert a degree measure into radians, we use the formula: \[ \text{Radian} = \text{Degree} \times \frac{\pi}{180^\circ} \] Given: \[ 135^\circ \] Conversion into Radians \[ 135^\circ \times \frac{\pi}{180^\circ} \] \[ = \frac{135\pi}{180} \] \[ = \frac{3\pi}{4} \] Answer \[ \frac{3\pi}{4}