In this discussion, we are going to prove the Co-Function Trig Identities using a right triangle.
Here is an outline of this discussion:
Proving sin(theta) = cos(pi/2 – theta)
Proving cos(theta) = sin(pi/2 – theta)
Proving tan(theta) = cot(pi/2 – theta)
Proving cot(theta) = tan(pi/2 – theta)
Proving csc(theta) = sec(pi/2 – theta)
Proving sec(theta) = csc(pi/2 – theta)
We begin with a right triangle.
We know that one of our angles are 90 degrees since we have a right triangle.
Let theta be an angle other than the 90 degree angle. Then the third angle must be (90 degrees – theta) since all angles within a triangle add up to 180 degrees.
Let “a” be the side opposite to theta, let “b” be the side adjacent to theta, and let “c” be the hypotenuse.
Proving sin(theta) = cos(pi/2 – theta)
We know from right triangle trigonometry that:
sin(theta) =a/c
and
cos(90 degrees – theta) = a/c.
Thus,
sin(theta) = cos (90 degrees – theta)
or
sin(theta) = cos(pi/2 – theta)
Proving cos(theta) = sin(pi/2 – theta)
cos(theta) = b/c
and
sin(90 degrees – theta) = b/c
Thus,
cos(theta) = sin(90 degrees – theta)
or
cos(theta) = sin(pi/2-theta)
Proving tan(theta) = cot(pi/2 – theta)
We know that by the quotient identity,
tan(theta) = sin(theta)/cos(theta)
We can use
sin(theta) = cos(90 degrees – theta)
and
cos(theta) = sin(90 degrees – theta).
So,
tan(theta)
=
sin(theta)/cos(theta)
=
cos(90 degrees – theta)/ sin(90 degrees – theta)
=
cot(90 degrees – theta)
Thus,
tan(theta) = cot(90 degrees – theta)
or
tan(theta) = cot(pi/2 – theta).
Proving cot(theta) = tan(pi/2 – theta)
We know that
cot(theta) = cos(theta)/sin(theta).
We can use
cos(theta) = sin(90 degrees – theta)
and
sin(theta) = cos(90 degrees – theta).
So,
cot(theta)
=
cos(theta)/sin(theta)
=
sin(90 degrees- theta)/cos(90 degree – theta)
=
tan(90 degrees – theta)
Thus,
cot(theta) = tan(90 degrees – theta)
or
cot(theta) = tan(pi/2-theta).
Proving csc(theta) = sec(pi/2 – theta)
W know that
csc(theta)= 1/sin(theta) and sec(theta) = 1/cos(theta)
So,
csc(theta)
=
1/sin(theta)
=
1/cos(90 degrees – theta)
=
sec(90 degrees – theta).
Thus,
csc(theta) = sec(90 degrees – theta)
or
csc(theta) = sec(pi/2 – theta)
Proving sec(theta) = csc(pi/2 – theta)
Similarly,
sec(theta)
=
1/cos(theta)
=
1/sin(90 degrees – theta)
=
csc(90 degrees – theta).
Thus
sec(theta) = csc(90 degrees – theta)
or
sec(theta) = csc(pi/2 – theta).
We have proven the co-function identities!
Thank you for reading! I hope that I can assist you on your math journey.
Sincerely,
➕MathAngel369➕
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