T-ratios of Complementary Angles

Two angles are called complementary iff the sum of their measures is 90° (or /2 radians)
    
sin (90° - ) = cos , cos (90° - ) = sin ,
tan (90° - ) = cot , cot (90° - ) = tan ,
sec (90° - ) = cosec , cosec (90° - ) = sec .

Illustrative Examples

Example

Without using trigonometric tables, evaluate
(i) sin23°/cos67°         (ii) tan65°/cot25°
(iii) sin 18° -cos 72°

Solution

(i) sin 23°/cos 67° = sin 23°/cos(90°-23°)
      = sin 23°/sin23°      (because cos (90° - ) = sin )
      = 1
(ii) tan 65°/cot25° = tan (90° -25°)/cot25°
      = cot25°/cot25°      (because tan (90° - ) = cot )
     = 1
(iii) sin 18° -cos 72° = sin 18° -cos (90° -18°)
      = sin 18° -sin 18° = 0

Exercise

Without using trigonometric tables, evaluate the following (1-2):

1. (i) cos18°/sin72°         (ii) cosec 31°/sec 59°
   (iii) cosec17°30'/sec 72°30'
2. (i) sin 62° -cos 28°      (ii) cosec 35° -sec 55°
   (iii) sin 35° sin 55° -cos 35° cos 55°
3. Express each of the following in terms of t-ratios of angles between 0° and 45°.
   (i) tan 81° +cos 72°     (ii) cot 49° +cosec 87°

Answers

1. (i) 1          (ii) 1       (iii) 1
2. (i) 0          (ii) 0       (iii) 0
3. (i) cot 9° +sin 18°    (ii) tan 41° +sec 3°