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.
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.
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.
Worked-out problems on trigonometrical ratios of complementary angles:
1. Without using trigonometric tables, evaluate tan65°cot25°tan65°cot25°
= tan65°tan65°tan65°tan65°, [Since cot (90° - θ) = tan θ]