Trigonometrical Ratios Of Complementary Angles

 

If the sum of two angles is one right angle or 90°, then one angle is said to be complementary of the other. Thus, 25° and 65°; θ° and (90 - θ)° are complementary to each other.

Suppose a rotating line rotates about O in the anti-clockwise sense and starting from its initial position

OX−→−OX→ traces out angle ∠XOY = θ, where θ is acute.

Take a point P on OY−→−OY→  and draw PQ¯¯¯¯¯¯¯¯PQ¯  perpendicular to OX.  Let, ∠OPQ = α. Then, we have,

α + θ = 90°

or, α = 90° -  θ.

Therefore, θ and α are complementary to each other.

Now, by the definition of trigonometric ratio,

From (i) and (iv) we have,

sin α = cos θ   

or,  sin (90° -  θ) = cos θ;

 

From (ii) and (v) we have,

cos α = sin θ   

or, cos (90° -  θ) = sin θ;

 

From (iii) and (vi) we have,

And tan α = 1/tan θ

or, tan (90° - θ) = cot θ.

 

Similarly, csc (90° - θ) = sec θ;

sec (90° - θ) = csc θ

and cot (90° - θ) = tan θ.

 

Therefore,

Sine of any angle      =  cosine of its complementary angle;

Cosine of any angle  =  sine of its complementary angle;

Tangent of any angle =  cotangent of its complementary angle.

 

Corollary:

Complementary Angles: Two angles are said to be complementary if their sum is 90°. Thus θ and (90° - θ) are complementary angles.

(i) sin (90° -  θ) = cos θ

(iii) tan (90° -  θ) = cot θ

(v) sec (90° -  θ) = csc θ

(ii) cos (90° -  θ) = sin θ

(iv) cot (90° -  θ) = tan θ

(vi) csc (90° -  θ)  = sec θ

We know there are six trigonometrical ratios in trigonometry. The above explanation will help us to find the trigonometrical ratios of complementary angles.

Example:

Worked-out problems on trigonometrical ratios of complementary angles:

1. Without using trigonometric tables, evaluate tan65°cot25°tan65°cot25°

Solution:

tan65°cot25°tan65°cot25°

= tan65°cot(90°−65°)tan65°cot(90°−65°)

=  tan65°tan65°tan65°tan65°, [Since cot (90° -  θ) = tan θ]

= 1