NEGATIVE AND COMPLEMENTARY ANGLE FORMULAE IN TRIGONOMETRY

Simplify:

Problem 1:

sin θ + sin (-θ)

Solution :

= sin θ + sin (-θ)

= sin θ - sin θ

= 0

Problem 2 :

tan (-θ) - tan θ

Solution :

= tan (-θ) - tan θ

= - tan θ - tan θ

= -2 tan θ

Problem 3 :

2 cos θ + cos (-θ)

Solution :

= 2 cos θ + cos (-θ)

= 2 cos θ + cos θ

= 3 cos θ

Problem 4 :

3 sin θ - sin (-θ)

Solution :

= 3 sin θ - sin (-θ)

= 3 sin θ + sin θ

= 4 sin θ

Problem 5 :

cos² (-α)

Solution:

= cos² (-α)

= cos² α

Problem 6 :

sin² (-α)

Solution :

= sin² (-α)

= (- sin α)²

= sin² α

Problem 7 :

cos (-α) cos α - sin (-α) sin α

Solution :

= cos (-α) cos α - sin (-α) sin α

= cos α cos α - (-sin α sin α)

= cos α cos α + sin α sin α

= cos² α + sin² α

= 1

Problem 8 :

2 sin θ - cos (90˚ - θ)

Solution :

= 2 sin θ - cos (90˚ - θ)

= 2 sin θ - sin θ

= sin θ

Problem 9 :

sin (-θ) - cos (90˚ - θ)

Solution :

= sin (-θ) - cos (90˚ - θ)

= - sin θ - cos (90˚ - θ)

= - sin θ - sin θ

= -2 sin θ

Problem 10 :

sin (90˚ - θ) - cos θ

Solution :

= sin (90˚ - θ) - cos θ

= cos θ - cos θ

= 0

Problem 11 :

3 cos (-θ) - 4 sin (π/2 - θ)

Solution :

= 3 cos (-θ) - 4 sin (π/2 - θ)

= 3 cos (θ) - 4 sin (π/2 - θ)

= 3 cos θ - 4 cos θ

= - cos θ

Problem 12 :

3 cos θ + sin (π/2 - θ)

Solution :

= 3 cos θ + sin (π/2 - θ)

= 3 cos θ + cos θ

= 4 cos θ

Problem 13 :

cos (π/2 - θ) + 4 sin θ

Solution :

= cos (π/2 - θ) + 4 sin θ

= sin θ + 4 sin θ

= 5 sin θ

Problem 14 :

Explain why sin (θ - ɸ) = -sin (ɸ - θ), cos (θ - ɸ) = cos (ɸ - θ)

Solution :

sin (θ - ɸ) = -sin (ɸ - θ)

(θ - ɸ) = - (θ - ɸ)

(θ - ɸ) = (ɸ - θ)

From (θ - ɸ) factoring negative sign. So, we will get (ɸ - θ).

cos (θ - ɸ) = cos (ɸ - θ)

Here, cos θ = cos (-θ)

So,

cos (θ - ɸ) = cos (- θ + ɸ)

cos (θ - ɸ) = cos (ɸ - θ)

Problem 15 :

Simplify:

sin θ / cos θ

Solution :

= sin θ / cos θ

= tan θ

Problem 16 :

sin (-θ) / cos (-θ)

Solution :

= sin (-θ) / cos (-θ)

= - sin θ / cos θ

= - tan θ

Problem 17 :

sin (π/2 - θ) / cos θ

Solution :

= sin (π/2 - θ) / cos θ

= cos θ / cos θ

= 1

Problem 18 :

-sin (-θ) / cos θ

Solution :

= -sin (-θ) / cos θ

= - (- sin θ) / cos θ

= sin θ / cos θ

= tan θ

Problem 19 :

cos (π/2 - θ) / sin (π/2 - θ)

Solution :

= cos (π/2 - θ) / sin (π/2 - θ)

= sin θ / cos θ

= tan θ

Problem 20 :

cos (π/2 - θ) / cos θ

Solution :

= cos (π/2 - θ) / cos θ

= sin θ / cos θ

= tan θ