Quotient identities
The quotient relation of two trigonometric
functions with each other is called quotient trigonometric identity or they are
simply called as quotient identity.
The quotient relations of trigonometric
functions are known as quotient identities.
There are
two quotient trigonometric identities in trigonometry.
Sine(Sin) and Cosine(Cos) with
Tangent(Tan)
The quotient
of sine function divided by cosine function at an angle θ is equal to tangent function at the same angle θ.
SinӨ
|
=
|
TanӨ
|
CosӨ
|
Cosine(Cos)
and Sine(Sin) with Cotangent(Cot)
The quotient
of cosine function by sine function at an angle is equal to cotangent function
at the same angle.
CosӨ
|
=
|
CotӨ
|
SinӨ
|
Proof of:
SinӨ
|
=
|
TanӨ
|
CosӨ
|
Quotient
identity of Sine and Cosine with Tangent
·
The quotient relation of sine and cosine
functions with tangent function is called the quotient identity of sine and
cosine functions with tangent function.
·
Formula
| SinӨ | = | TanӨ |
| CosӨ |
Sine
function can be divided by the cosine function and the quotient of them
represents another trigonometric function tangent. The quotient relation of
sine and cosine functions with tangent function is used as basic trigonometric
formula in mathematics.
ΔBAC is a right angled triangle, whose angle is assumed
as theta (θ).
Define Sine and Cosine functions in terms of sides of the
right angled triangle
Express sine and cosine functions in terms of ratio of
the sides of the right angled triangle BAC.
SinӨ
|
=
|
Length
of Opposite side
|
=
|
BC
|
|||
Length
of Hypotenuse
|
AC
|
||||||
CosӨ
|
=
|
Length of Adjacent side
|
=
|
AB
|
|||
Length of Hypotenuse
|
AC
|
||||||
Divide Sine function
by Cosine
Now dividing
sine function by cosine function and simplifying it to obtain quotient of sin &
Cos.
SinӨ
|
=
|
BC
|
AC
|
||
CosӨ
|
AB
|
|
AC
|
SinӨ
|
=
|
|||
BC
|
X
|
AC
|
||
CosӨ
|
AC
|
AB
|
||
By
dividing AC/AC=1 , We get,
SinӨ
|
=
|
|
BC
|
||
CosӨ
|
AB
|
|
Expressing quotient of TanӨ in terms of the sides of right angled
triangle
The quotient of sine by cosine function is the ratio of
two sides. Actually, the ratio of length of opposite side (BC) divided by
the length of adjacent side (AB) which represents tangent function ,
TanӨ
|
=
|
Length of Opposite side
|
=
|
BC
|
Length of Adjacent side
|
AB
|
Therefore, it is proved that the quotient of sine
function by cosine function is equal to the tangent function as proved above,
∴
SinӨ
|
=
|
TanӨ
|
CosӨ
|
·
Similarly Cosine and Sine
with Cotangent
CosӨ
|
=
|
CotӨ
|
SinӨ
|
||
Can be proved.