Basic trigonometric identities

Here we will show you the basic trigonometric identities. They are to be used only for triangles that have a right angle. If the triangle is not a right triangle, none of the formulas shown below are valid. It is extremely important to keep this in mind. A triangle is a right triangle if one of the angles is equal to 90 degrees.

First make sure you examine the diagram below so you understand what the opposite and adjacent sides are.




Notice that the hypotenuse is the line that is neither vertical nor horizontal.

In this case, we did not put any label on the triangle to identify the hypotenuse. However, it is the line that is slanted.

Remember an angle is made with the sides. The side that is opposite to the angle Θ is the side that is not used to create the angle. I hope this helps to identify the opposite side.

The adjacent side is one of the sides that is used to make the angle, but it is not the hypotenuse or the slanted side.

The adjacent side is going to be either the horizontal side or the vertical side.



Sine function


sin Θ  =  
Opposite side / Hypotenuse


sin Θ1  =  
Opposite side / Hypotenuse
sin Θ  =  
Leg 2 / H


sin Θ1  =  
Leg 1 / H


Cosine function


cos Θ  =  
Adjacent side / Hypotenuse


cos Θ1  =  
Adjacent side / Hypotenuse
cos Θ  =  
Leg 1 / H


cos Θ1  =  
Leg 2 / H


Tangent function


tan Θ  =  
Opposite side / Adjacent


tan Θ1  =  
Opposite side / Adjacent
tan Θ  =  
Leg 2 / Leg 1


tan Θ1  =  
Leg 1 / Leg 2


Cotangent function


cotan Θ  =  
Adjacent side / Opposite side


cotan Θ1  =  
Adjacent side / Opposite side
cotan Θ  =  
Leg 1 / Leg 2


cotan Θ1  =  
Leg 2 / Leg 1


 Secant function


sec Θ  =  
Hypotenuse / Adjacent side


sec Θ1  =  
Hypotenuse side / Adjacent side
sec Θ  =  
H / Leg 1


sec Θ1  =  
H / Leg 2


Cosecant  function


csc Θ  =  
Hypotenuse / Opposite side


csc Θ1  =  
Hypotenuse / Opposite side
csc Θ  =  
H / Leg 2


csc Θ1  =  
H / Leg 1



More trigonometric identities

The tangent is the ratio of the sine to the cosine.

The cotangent is the inverse of the tangent.

The cosecant is the inverse of the sine

The secant is the inverse of the cosine

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