Trigonometric Ratios of Complementary Angles

Ratios of Complementary Angles (Complementary Formulae of Trigonometric Ratios of Angles)

It is known from Geometrical Mathematics concept that two angles are said to be complementary angles if and only if their sum is 90° and each angle is said to be the complement of the other angle. For example :- 62° and 28° are complementary angles because (62° + 28° = 90°) & angle complement to 62° is 28° as well as angle complement to 28° is 62°.


Let us understand the details of Trigonometric Ratios mentioned as under.

Let us consider a Right-angled triangle, △PQR right angled at Q that is ∠PQR = 90°

Ratios of Complementary Angles

In this figure, we have some specific observations:

In △PQR, we have

  • For angle θ, PQ is the BASE and QR is the PERPENDICULAR (Opposite Side)
  • For angle 90°–θ, QR is the BASE and PQ is the PERPENDICULAR (Opposite Side)
  • PR is the HYPOTENUSE (It is longest side of any right angled triangle)


Also, in △PQR, let us suppose that ∠QPR = θ. Then in accordance with Angle Sum Property of a Triangle, we know that the sum of three angles of Triangle is always 180° and so, from △PQR, we have

∠PQR + ∠QPR + QRP = 180°

⇒ 90° + θ + QRP = 180°

QRP = 180° – 90° – θ

QRP = 90° – θ

Ratios of Complementary Angles

So, from the above expression, we can derive 6(six) Complementary Formulae of Trigonometric Ratios of Angles mentioned as under:


  • From (a) and (ii), we have

           sin (90°–θ) = cos θ

  • From (b) and (i), we have

          cos (90°–θ) = sin θ

  • From (c) and (iv), we have

         tan (90°–θ) = cot θ

  • From (d) and (iii), we have

          cot (90°–θ) = tan θ

  • From (e) and (vi), we have

          sec (90°–θ) = cosec θ

  • From (f) and (v), we have

          cosec (90°–θ) = sec θ




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