CONVERTING DEGREES TO RADIANS AND RADIANS TO DEGREES

Example 1 :

Convert each of the following degree measures to radian measures. 

(i) 30° (ii) 135° (iii) -205° (iv) 150° (v) 330°

Solution :

(i) 30° :

30° = 30° x (π/180°) radians

   = π/6 radians

(ii) 135° :

135° = 135° x (π/180°) radians 

= 3π/4 radians

(iii) -205° :

-205° = -205° x (π/180°) radians

= -41π/36 radians

(iii) 150° :

150° = 150° x (π/180°) radians

= 5π/6 radians

(v) 330° :

330° = 330° x (π/180°) radians

   = (11π/6) radians

Example 2 :

Convert each of the following radian measures to degree measures. 

(i) π/3 (ii) π/9 (iii) 2π/5 (iv) 7π/3 (v) 10π/9 (i) π/3

Solution :

(i) π/3 :

π/3 = π/3 x (180°/π)

= 60°

(ii) π/9 :

π/9 = π/9 x (180°/π)

= 20°

(iii) 2π/5 :

2π/5 = 2π/5 x (180°/π)

= 72°

(iv) 7π/3 :

70π/3 = 7π/3 x (180°/π)

= 420°

(v) 10π/9 :

10π/9 = 10π/9 x (180°/π)

= 200°

Example 3 :

Convert as indicated.

a. 29.68° to DMS form

b. 46°18′21′′ to decimal degree form

Solution :

a) 

1° = 60 minutes

1 minute = 60 seconds

29.68° = 29 + 0.68°

To convert 0.68° in to minute, we have to multiply by 60

= 29 + 0.68(60)

= 29 + 40.8

= 29 + 40 + 0.8

To convert 0.8 into seconds, we have to multiply by 60

= 29 + 40 + 0.8 x 60

= 29°  40' + 48''

= 29°  40' 48''

b. 46°18′21′′ to decimal degree form

46°18′21′′ = 46° + 18′ + 21′′

To convert minutes to degree, we have to divide by 60

= 48 + 18 x (1/60)

= 48 + 0.3

= 48.3

To convert seconds to degree, we have to divide by (60 x 60).

= 48.3 + 21 x (1/3600)

= 48.3 + 0.0058

= 48.3058°

Example 4 :

Adding and Subtracting Angles in DMS Form Perform the indicated operations.

a. 36° 58′ 21′′ + 5° 06′ 45′′

b. 36° 17′ − 15° 46′ 15′′

Solution :

a. 36° 58′ 21′′ + 5° 06′ 45′′

Converting degree, minutes and seconds separately.

= (36 + 5) + (58 + 06) + (21 + 45)

= 41 + 64 + 66

= 41 + (60 + 4 minutes) + (60 + 6 seconds)

= 41° + (1 degree + 4 minutes) + (1 minute + 6 seconds)

= 42° 05' 06''

b. 36° 17′ − 15° 46′ 15′′

Converting degree, minutes and seconds separately.

= (36 - 15) + (17 - 46) + (0 - 15)

Borrowing 1° and 1 minute

= (35 - 15) + (60 + 17 - 46) + (0 - 15)

= 20 + 31 + (0 - 15)

= 20 + 30 + (60 - 15)

= 20° 30' 45''

Example 5 :

Evaluate each trigonometric expressions :

a)  cos (4π/3)

b) tan (-210)

c) csc (11π/4)

Solution :

a)  cos (4π/3)

The given angle 4π/3 lies in 3rd quadrant. Reference angle formula will be θ - π. Using ASTC,  cos (4π/3) will be negative.

Reference angle = (4π/3) - π

= π/3

cos (4π/3) = -cos π/3

= -1/2

b) tan (-210)

= - tan 210

210 lies in 3rd quadrant. Using ASTC, value of tan 210 will be negative.

= tan 210

= tan (210 - 180)

= tan 30

= 1/√3

c) csc (11π/4)

= csc (11π/4)

= csc (11π/4 - 2π)

= csc (3π/4)

= csc (π - π/4)

(11π/4) lies in 2nd quadrant. Using ASTC, value of csc (π/4) will be positive.

= csc (π/4)

= √2

Example 6 :

sin 420 cos 390 + cos (-300) sin (-330) = 1

Solution :

sin 420 cos 390 + cos (-300) sin (-330) = 1

sin 420 = sin (360+ 60)

= sin 60

= √3/2

cos 390 = cos (360 + 30)

= cos 30

= √3/2

cos (-300) = cos 300

= cos (270 + 30)

= sin 30

= 1/2

sin (-330) = - sin 330

= - sin (360 - 30)

= sin 30

= 1/2

= (√3/2) √3/2 + 1/2 (1/2)

= 3/4 + 1/14

= 4/4

= 1

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