Trigonometric Functions - Relation between degree and radian

Convert $40^{\circ} 20'$ into radian measure.

$40^{\circ} 20' = 40 \frac{20}{60}^{\circ} = 40 \frac{1}{3}^{\circ} = \frac{121}{3}^{\circ}$. Since $180^{\circ} = \pi$ radians, then $\frac{121}{3}^{\circ} = \frac{\pi}{180} \times \frac{121}{3} = \frac{121\pi}{540}$ radians.

First, convert the minutes into a fraction of a degree. Then, use the conversion formula by multiplying the total degrees by $\frac{\pi}{180}$.

Find the degree measure of $6$ radians (use $\pi = \frac{22}{7}$).

$\text{Degree measure} = \frac{180}{\pi} \times 6 = \frac{180 \times 7 \times 6}{22} = \frac{3780}{11} = 343 \frac{7}{11}^{\circ}$. To convert the fraction: $\frac{7}{11}^{\circ} = \frac{7 \times 60}{11}' = \frac{420}{11}' = 38 \frac{2}{11}'$. Then $\frac{2}{11}' = \frac{2 \times 60}{11}'' \approx 11''$. Final answer: $343^{\circ} 38' 11''$.

Multiply radians by $\frac{180}{\pi}$. Convert the fractional part of the degree into minutes by multiplying by $60$, and the fractional part of the minute into seconds by multiplying by $60$ again.

Find the radius of the circle in which a central angle of $60^{\circ}$ intercepts an arc of length $37.4$ cm (use $\pi = \frac{22}{7}$).

Given $l = 37.4$ and $\theta = 60^{\circ}$. Convert $\theta$ to radians: $\theta = 60 \times \frac{\pi}{180} = \frac{\pi}{3}$ radians. Using $r = \frac{l}{\theta}$: $r = \frac{37.4}{\pi / 3} = \frac{37.4 \times 3}{22/7} = \frac{37.4 \times 3 \times 7}{22} = 35.7 \text{ cm}$

The formula $\theta = \frac{l}{r}$ only works when $\theta$ is in radians. First convert $60^{\circ}$ to radians before calculating $r$.

$\text{Number of revolutions in 60 seconds} = 360$ $\text{Number of revolutions in 1 second} = \frac{360}{60} = 6$ $\text{Angle turned in 1 revolution} = 2\pi \text{ radians}$ $\text{Angle turned in 6 revolutions} = 6 \times 2\pi = 12\pi \text{ radians}$

First, find the number of revolutions per second. Since one full revolution equals $2\pi$ radians, multiply the number of revolutions by $2\pi$ to find the total radian measure.

$\text{Given: } r = 100 \text{ cm}, l = 22 \text{ cm}$ $\theta = \frac{l}{r} = \frac{22}{100} \text{ radians}$ $\text{Degree measure} = \frac{180}{\pi} \times \theta$ $\text{Degree measure} = \frac{180}{22/7} \times \frac{22}{100}$ $\text{Degree measure} = \frac{180 \times 7 \times 22}{22 \times 100} = \frac{126}{10} = 12.6^{\circ}$ $\text{To convert } 0.6^{\circ} \text{ to minutes: } 0.6 \times 60 = 36'$ $\text{Result: } 12^{\circ} 36'$

Use the formula $\theta = l/r$ to find the angle in radians. Then convert the radians to degrees by multiplying with $180/\pi$. Finally, convert the decimal part of the degrees into minutes.