Proving the the Reciprocal and Quotient Identities (MathAngel369)
In this discussion, we are going to prove the reciprocal and Quotient Identities using a right triangle.
Here is an outline of this discussion:
Review of Right a Triangle Trigonometry
The reciprocal Identities for Sine and Cosecant
The Reciprocal Identities for Cosine and Secant
The Reciprocal Identities for Tangent and Cotangent
The Quotient Identity for Tangent
The Quotient Identity for Cotangent
We begin with what we know. Recall a right triangle.
Review of Right Triangle Trigonometry
Given a right triangle and an angle theta, let
O= the side opposite of the given angle
A = the side adjacent to the given angle
H = the hypotenuse
We know from Right-Triangle Trigonometry that:
sin(theta) = O/H
cos(theta) = A/H
tan(theta) = O/A
cot(theta) = A/O
csc(theta) = H/O
sec(theta) = H/A
The Reciprocal Identity for Sine and Cosecant
Take a closer look at sine and cosecant:
sin(theta) = O/H
csc(theta) = H/O
Clearly, we see that sine and cosecant are reciprocals of each other.
Let’s verify algebraically:
1/sin(theta) = 1/(O/H) = H/O = csc(theta)
Thus, csc(theta) = 1/sin(theta)
1/csc(theta) = 1/(H/O) = O/H = sin(theta)
Thus, sin(theta) = 1/csc(theta)
The Reciprocal Identity for Cosine and Secant
Now take a look at cosine and secant:
cos(theta) = A/H
sec(theta) = H/A
We see that cosine and secant are reciprocals of each other.
Let’s verify algebraically:
1/cos(theta) = 1/(A/H) = H/A = sec(theta)
Thus, sec(theta) = 1/cos(theta)
1/sec(theta) = 1/(H/A) = A/H = cos(theta)
Thus, cos(theta) = 1/sec(theta)
The Reciprocal Identity for Tangent and Cotangent
Last, take a look at tangent and cotangent.
tan(theta) = O/A
cot(theta) = A/O
We see that tangent and cotangent are reciprocals of each other.
Verifying algebraically:
1/tan(theta) = 1/(O/A) = A/O = cot(theta)
Thus, cot(theta) = 1/tan(theta)
1/cot(theta) = 1/(A/O) = O/A = tan(theta)
Thus, tan(theta) = 1/cot(theta).
The Quotient Identity for Tangent
To for the quotient identity for tangent, let us look at sine and cosine. We know that
sin(theta)=O/H
and
cos(theta) = A/H
So,
sin(theta)/cos(theta)
=
(O/H)/(A/H)
=
(O/H)*(H/A)
=
O/A
=
tan(theta)
Thus,
tan(theta) = O/H
The Quotient Identity for Cotangent
To find the quotient identity for cotangent, we use sine and cosine again.
So,
cos(theta)/sin(theta)
=
(A/H)/(O/H)
=
(A/H)*(H/O)
=
A/O
=
cot(theta)
Thus,
cot(theta) = A/O
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