Introduction to Trig Ratios
In the adjoining figure, we have Δ ABC right angled at C.
• Sine θ
• Cosine θ
• Cosecant θ
• Secant θ
• Cotangent θ
Let the Hypotenuse in Δ ABC = h
Adjacent in Δ ABC = b
Opposite in Δ ABC = a
Now, we define the above mentioned trigonometric ratios:
• Sine θ or Sin θ = Opposite / Hypotenuse = a / h
• Cosine θ or Cos θ = Adjacent / Hypotenuse = b / h
• Tangent θ or Tan θ = Opposite / Adjacent = a / b
• Cosecant θ or Cosec θ = Hypotenuse / Opposite = h / a
• Secant θ or Sec θ = Hypotenuse / Adjacent = h / b
• Cotangent θ or Cot θ = Adjacent / Opposite = b / a
From the above discussion, it is clear that the last three trigonometric ratios are opposite of the first three trigonometric ratios respectively.
That is,
• Cosecant θ or Cosec θ = 1 / Sine θ
• Secant θ or Sec θ = 1 / Cosine θ
• Cotangent θ or Cot θ = 1 / Tangent θ
There is one short method for remembering all six trigonometric ratios.
SOH
‘S’ stands for Sine
‘O’ stands for Opposite
‘H’ stands for Hypotenuse
Sine = Opposite / Hypotenuse
CAH
‘C’ stands for Cosine
‘A’ stands for Adjacent
‘H’ stands for Hypotenuse
Cosine = Adjacent / Hypotenuse
TOA
‘T’ stands for Tangent
‘O’ stands for Opposite
‘A’ stands for Adjacent
Tangent = Opposite / Adjacent
As discussed above, that Cosecant, Secant, and Cotangent are opposites of Sine, Cosine and Tangent respectively.
Let’s solve few problems based on the above discussion:
In Δ ABC, right angled at A, if AB = 12, AC = 5 and BC = 13, find all the six trigonometric ratios of angle B.
With reference to above Δ ABC we have,
Opposite = AC = 5
Adjacent = AB = 12
Hypotenuse = BC = 13
Using the definitions of trigonometric ratios, we have
Sine B = Opposite / Hypotenuse = AC / BC = 5 / 13
Cosine B = Adjacent / Hypotenuse = AB / BC = 12 / 13
Tangent B = Opposite / Adjacent = AC / AB = 5 / 12
Cosecant B = Hypotenuse / Opposite = BC / AC = 13 / 5
Secant B = Hypotenuse / Adjacent = BC / AB = 13 / 12
Cotangent B = Adjacent / Opposite = AB / AC = 12 / 5
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