Trigonometry Proving Pythagoras’ Theorem

Trigonometry Proving Pythagoras’ Theorem

(Trigonometry Proving Pythagoras’ Theorem)

Can trigonometry be used to prove the Pythagorean theorem? Please provide an explanation and examples for your answer following MLA Guidelines.

Response.

Yes, trigonometry can be used to prove the Pythagorean theorem. This approach relies on fundamental trigonometric identities and the concept of similar triangles. Here’s an explanation and example:

Explanation

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse ($c$) is equal to the sum of the squares of the other two sides ($a$ and $b$):
a2+b2=c2.a^2 + b^2 = c^2.a2+b2=c2.

Using trigonometry, this theorem can be demonstrated by analyzing the relationships between the sides of the triangle and the angles within it. Specifically, trigonometric functions like sine ($\sin$) and cosine ($\cos$) are defined in terms of the sides of a right triangle. These functions help establish the relationship between the sides, allowing for a trigonometric proof.

Example: Trigonometric Proof of the Pythagorean Theorem

1. Consider a right triangle:
   • Let the angle opposite side $a$ be $\theta$.
   • By definition, sin⁡(θ)=ac\sin(\theta) = \frac{a}{c}sin(θ)=ca​ and cos⁡(θ)=bc\cos(\theta) = \frac{b}{c}cos(θ)=cb​.
2. Express the sides in terms of $c$:
   • $a=c\cdot \sin (\theta )$
   • $b=c\cdot \cos (\theta )$
3. Square both equations:
   • a2=c2⋅sin⁡2(θ)a^2 = c^2 \cdot \sin^2(\theta)a2=c2⋅sin2(θ)
   • b2=c2⋅cos⁡2(θ)b^2 = c^2 \cdot \cos^2(\theta)b2=c2⋅cos2(θ)
4. Add the equations together:a2+b2=c2⋅sin⁡2(θ)+c2⋅cos⁡2(θ)a^2 + b^2 = c^2 \cdot \sin^2(\theta) + c^2 \cdot \cos^2(\theta)a2+b2=c2⋅sin2(θ)+c2⋅cos2(θ)
5. Factor out c2c^2c2:a2+b2=c2(sin⁡2(θ)+cos⁡2(θ))a^2 + b^2 = c^2 (\sin^2(\theta) + \cos^2(\theta))a2+b2=c2(sin2(θ)+cos2(θ))
6. Use the Pythagorean identity (sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1sin2(θ)+cos2(θ)=1):a2+b2=c2⋅1a^2 + b^2 = c^2 \cdot 1a2+b2=c2⋅1
7. Simplify:a2+b2=c2a^2 + b^2 = c^2a2+b2=c2

This completes the trigonometric proof of the Pythagorean theorem.

Works Cited

Burton, David M. Elementary Number Theory. 7th ed., McGraw-Hill, 2011.
Simmons, George F. Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry. 3rd ed., W.W. Norton & Company, 2003.

 

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