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# Activity 3 Verify the above result by taking the circles having different radii. Let an angle have its measure in radian and in degrees. Then its proportion with the straight angle is the same in either measure. - $\frac{\theta}{180°}$ = $\frac{\phi}{180}$ We use this relation to convert radians...

# Activity 3 Verify the above result by taking the circles having different radii. Let an angle have its measure in radian and in degrees. Then its proportion with the straight angle is the same in either measure. - $\frac{\theta}{180°}$ = $\frac{\phi}{180}$ We use this relation to convert radians into degrees and vice-versa. ## Notes 1. To convert degrees into radians, multiply the degree measure by $\frac{\pi}{180}$. 2. To convert radians into degrees, multiply the radian measure by $\frac{180}{\pi}$. 3. Taking $\pi$ = 3.14, we have 1° = $\frac{180}{\pi}$ = 57.3248° Here the fractional degree is given in decimal fractions. It can be converted into minutes and seconds as follows: - 0.3248° = (0.3248 x 60)' - = 19.488' - = 19' + (0.488 x 60)" - = 19' + 29" - Thus, 1° = 57° 19' 29" 4. In the table given below, certain degree measures are expressed in terms of radians: | Degree | 15 | 30 | 45 | 60 | 90 | 120 | 180 | 270 | 360 | |---|---|---|---|---|---|---|---|---|---| | Radian | $\frac{\pi}{12}$ | $\frac{\pi}{6}$ | $\frac{\pi}{4}$ | $\frac{\pi}{3}$ | $\frac{\pi}{2}$ | $\frac{2\pi}{3}$ | $\pi$ | $\frac{3\pi}{2}$ | $2\pi$ | ## Relation Between Angle And Time In a clock, R is a rotation: | Min Hand | Hr Hand | |---|---| | 1R = 360° | 1R = 360° | | 1R = 60 min | 1R = 12 Hrs | | 60 min = 360° | 12 Hrs = 360° | | 1 min = 6° rotation | 1 Hr = 30° | | 60 min = 30° | 1 min = $\frac{1}{2}$ | | | | The word "minute" is used for time measurement as well as 60° part of a degree of angle. Please note that "minute" in time and "minute" as a fraction of degree angle are different. # Solved Examples ### Example 1 Convert the following degree measures into radians. 1. 70° 2. -120° 3. $\frac{1}{4}$° 4. $(\frac{1}{4})^°$ ### Solution We know that 0° = $(\frac{\theta}{180}) \pi$: 1. 70° = $(\frac{70}{180})\pi$ = $\frac{7\pi}{18}$ 2. -120° = $(\frac{-120}{180})\pi$ = $\frac{-2\pi}{3}$ 3. $(\frac{1}{4})°$ = $(\frac{1}{4})(\frac{\pi}{180})$ = $\frac{\pi}{720}$