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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.

trigonometric relationships-1sin θ  =  {opposite}/{hypotenuse}~=~{AB}/{AC}

cos θ  =  {adjacent}/{hypotenuse}~=~{BC}/{AC}

tan θ  =  {opposite}/{adjacent}~=~{AB}/{BC}

 

Now we look at some other trigonometric relationships:

sin (90 – θ)  =  {opposite}/{hypotenuse}~=~{BC}/{AC} = cos θ

cos (90 – θ)  =  {adjacent}/{hypotenuse}~=~{AB}/{AC} = sin θ

Hence sin (90 – θ) = cos θ, and

cos (90 – θ) = sin θ

tan θ  =  {opposite}/{adjacent}~~=~~{{opposite}/{hypotenuse}}/{{adjacent}/{hypotenuse}}~~=~~{sin theta}/{cos theta}

 

We define three inverse trigonometric ratios as:

cosecant angle  =  cosec θ  =  1/{sin theta}

secant angle  =  sec θ  =  1/{cos theta}

cotangent angle  =  cot θ  =  1/{tan theta}

Hence cot θ  =  1/{tan theta}~=~1/({sin theta}/{cos theta})~=~{cos theta}/{sin theta}

Therefore tan (90 – θ)  =  BC/AB~~=~~{BC/AC}/{AB/AC}~~=~~{cos theta}/{sin theta}  =  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  =  1/{sin theta}~=~1/{cos~(90~-~theta)}  =  sec (90 – θ)

sec θ  =  1/{cos theta}~=~1/{sin~(90~-~theta)}  =  cosec (90 – θ)

cot θ  =  1/{tan theta}  =  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.