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# Trigonometric Relationships

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.