Key Points - Some Application to Trigonometry

- The height or length of an object or the distance between two distant objects can be determined with the help of trigonometric ratio.

- The line of sight is the line drawn from the eye of an observer to the point of the object viewed by the observer.

- Trigonometric Ratios:

In ?ABC, ∠B = 90o , for angle ' A '
Sin A = Perpendicular / Hypotenuse
Cos A =  Base / Hypotenuse
tan A = Perpendicular / Base 
Cot A = Base / Perpendicular
Sec A =  Hypotenuse / Base
cos ec A = Hypotenuse / Perpendicular

Reciprocal Relations: 
sin θ = 1/ cos ec θ
cos ec θ = 1 / sin θ
cos θ =  1 / sec θ
sec θ = 1 / cos θ
tan θ =  1/ cot θ
cot θ = 1 / tan θ

Quotient Relations: 
tan θ =   sin θ / cos θ
cot θ = cos θ / sin θ

Identities: 

sin^2 + cos^2 = 1 ⇒ sin^2 θ = 1 – cos^2θ and cos^2 θ = 1 – sin^2 θ

1 + cot^2 θ = cos ec2θ ⇒ cot 2 θ = cos ec2θ − 1 and cos ec2θ − cot^2 θ = 1
1 + tan^2 θ = sec2 θ ⇒ tan^2 θ = sec2 θ − 1 and sec2 θ − tan^2 θ = 1

- Trigonometric Ratios of Some Specific Angles:

- Trigonometric Ratios of Complementary Angles:
 sin (90o − θ) = cos θ , cos (90o − θ) = sin θ , tan (90o − θ) = cot θ cot (90o − θ) = tan θ , sec (90o − θ) = cos ec θ , cos ec (90o − θ) = sec θ

- Line of Sight: The line of sight is the line drawn from the eyes of an observer to a point in the object viewed by the observer.

- Angle of Elevation: The angle of elevation is the angle formed by the line of sight with the horizontal, when it is above the horizontal level i.e. the case when we raise our head to look at the object.

- Angle of Depression: The angle of depression is the angle formed by the line of sight with the horizontal when it is below the horizontal i.e. case when we lower our head to look at the object.