In this discussion, we are going to prove the Pythagorean Trig Identities.
Here is an outline of this article:
• Refresher on Previous Content
• Prove that sin^2(ϴ) + cos^2(ϴ) = 1
• Prove that 1 + cot^2(ϴ) = csc^2(ϴ)
• Prove that tan^2(ϴ) + 1 = sec^2(ϴ)
• Showing Similar Trig Identities
It is assumed that you already have an understanding of Pythagorean Theorem, Right Triangle Trigonometry, the Reciprocal Identities, and the Quotient Identities.
If you need a refresher, reflect on the content below, then continue.
Let’s begin!
Prove that sin^2(ϴ) + cos^2(ϴ) = 1
You are given a right triangle, and an angle, ϴ.
Let:
O = the opposite side of the angle, ϴ ;
A = the adjacent side of the angle, ϴ;
H = the hypotenuse.
We know that:
sin(ϴ)=O/Hi
cos(ϴ) = A/H
Solve for O and A :
O = Hsin(ϴ)
A = Hcos(ϴ)
Now, given a right triangle with sides a and b with hypotenuse c,
a^2+ b^2= c^2 by Pythagorean’s Theorem.
Using the above analogy, we know that
O^2 + A^2 = H^2
Plug O and A into the above equation:
[Hsin(ϴ)]^2 + [Hcos(ϴ)]^2 = H^2
Rewrite:
H^2sin^2(ϴ) + H^2cos^2(ϴ) = H^2
Now divide both side by H^2 to get your first identity:
sin^2(ϴ) + cos^2(ϴ) = 1
We will prove the next two identities using sin^2(ϴ) + cos^2(ϴ) = 1.
Prove that 1 + cot^2(ϴ) = csc^2(ϴ)
Begin with:
sin^2(ϴ) + cos^2(ϴ) = 1
Divide both sides of the equation by sin^2(ϴ).
sin^2(ϴ)/ sin^2(ϴ) + cos^2(ϴ)/ sin^2(ϴ) = 1/ sin^2(ϴ)
We know that
sin^2(ϴ)/ sin^2(ϴ) = 1
cos^2(ϴ)/ sin^2(ϴ) = cot^2(ϴ)
1/ sin^2(ϴ) = csc^2(ϴ)
So, we simplify to get your second identity:
1 + cot^2(ϴ) = csc^2(ϴ)
Prove that tan^2(ϴ) + 1 = sec^2(ϴ)
For the third identity, begin again with
sin^2(ϴ) + cos^2(ϴ) = 1
This time, divide both sides of the equation by cos2(ϴ).
sin^2(ϴ)/ cos^2(ϴ) + cos^2(ϴ)/ cos^2(ϴ) = 1/ cos^2(ϴ)
We know that
sin^2(ϴ) / cos^2(ϴ) = tan^2(ϴ)
cos^2(ϴ) / cos^2(ϴ) = 1
1/ cos^2(ϴ) = sec^2(ϴ)
So, we simplify to get the third Identity:
tan^2(ϴ) + 1 = sec^2(ϴ)
Showing Similar Trig Identities
We can come up with other identities using the three that we have proved:
sin^2(ϴ)+ cos^2(ϴ) = 1 can be re-written as:
sin^2(ϴ) = 1 – cos^2(ϴ)
cos^2(ϴ) = 1 – sin^2(ϴ)
1 + cot^2(ϴ) = csc^2(ϴ) can be re-written as:
cot^2(ϴ) = csc^2(ϴ) -1
csc^2(ϴ) – cot^2(ϴ) = 1
tan^2(ϴ) + 1 = sec^2(ϴ) can be re-written as:
tan^2(ϴ) = sec^2(ϴ) – 1
sec^2(ϴ) – tan^2(ϴ) = 1
Now you are ready to verify more challenging trig identities using the above trig identities. That will be in the next video, so stay tuned!
Thank you for reading!
I hope that I can assist you on your math journey!
Sincerely,
MathAngel369
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