TRIGONOMETRIC RATIOS FOR COMPLEMENTARY ANGLES EXAMPLES

Example 1 :

Find the value of the following.

(cos 47/sin 43)² + (sin 72/cos 18)² - 2cos² 45

Solution :

  =  (cos 47/sin 43)² + (sin 72/cos 18)² - 2cos² 45

cos 47  = sin (90 - 47)  =  sin 43

sin 72  =  cos (90 - 72)  =  cos 18

cos 45  =  1/√2

 =  (sin 43/sin 43)² + (cos 18/cos 18)² - 2 (1/√2)²

  =  1² + 1² - 1

 =  1

Example 2 :

Find the value of the following.

(cos 70/sin 20)² + (cos 59/sin 31)² + cos θ/sin(90-θ) - 8 cos² 60

L.H.S :

 =  (cos 70/sin 20)² + (cos 59/sin 31)² + cos θ/sin(90-θ) - 8 cos² 60

cos 70  =  sin (90 - 70)  =  sin 20

cos 59  =  sin (90 - 59)  =  sin 31

sin(90-θ)  =  cos θ

cos² 60  =  (1/2)²  =  1/4

= (sin 20/sin 20)² + (sin 31/sin 31)² + (cos θ/cosθ) - 8(1/4)

  =  1 + 1 + 1 - 2

  =  3 - 2

  =  1

Example 3 :

Find the value of the following.

tan 15° tan 30° tan 45° tan 60° tan 75°

Solution :

  =  tan 15° tan 30° tan 45° tan 60° tan 75°

tan 75°  =  cot (90 - 15)  =  cot 15

tan 30°  =  1/√3, tan 45°  =  1, tan 60°  =  √3

  =  tan 15° (1/√3) 1 (√3) cot 15°

  =  tan 15° (1/tan 15°)

  =  1

Example 4 :

Find the value of the following.

 [cot θ / tan (90 - θ)] + [cos (90 - θ) tan θ sec(90 - θ)/sin (90 - θ) cot (90 - θ) cosec (90 - θ)]

Solution :

=  [cot θ / tan (90 - θ)] + [cos (90 - θ) tan θ sec(90 - θ)/sin (90 - θ) cot (90 - θ) cosec (90 - θ)]

= [cot θ/cot θ] + [sin θ tan θ cosec θ/cos θ tan θ sec θ]

= 1 + (sin θ cosec θ/cos θ sec θ)

= 1 + 1

 = 2

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