Consider a right-angled triangle Δ ABC. We know the existing trigonometric relationships between the opposite, adjacent sides and the hypotenuse of the triangle.
sin θ =
cos θ =
tan θ =
Now we look at some other trigonometric relationships:
sin (90 – θ) = = cos θ
cos (90 – θ) = = sin θ
Hence sin (90 – θ) = cos θ, and
cos (90 – θ) = sin θ
tan θ =
We define three inverse trigonometric ratios as:
cosecant angle = cosec θ =
secant angle = sec θ =
cotangent angle = cot θ =
Hence cot θ =
Therefore tan (90 – θ) = = cot θ
Sine and cosine are called complementary ratios. Similarly, tangent and co-tangent are called complementary ratios.
Summary of Trigonometric Relationships
sin θ = cos (90 – θ)
cos θ = sin (90 – θ)
tan θ = cot (90 – θ)
cosec θ = = sec (90 – θ)
sec θ = = cosec (90 – θ)
cot θ = = tan (90 – θ)
Hence, in a right-angled triangle, θ and (90 – θ) are complementary angles. When two angles are complementary, we can find their complementary ratios given one of the angles.
Let us now look at some examples of trigonometric relationships.