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## Trigonometry

### Unit 2: Lesson 3

The Pythagorean identity

# Pythagorean identity review

Review the Pythagorean trigonometric identity and use it to solve problems.

## What is the Pythagorean identity?

sine, squared, left parenthesis, theta, right parenthesis, plus, cosine, squared, left parenthesis, theta, right parenthesis, equals, 1
This identity is true for all real values of theta. It is a result of applying the Pythagorean theorem on the right triangle that is formed in the unit circle for each theta.

## What problems can I solve with the Pythagorean identity?

Like any identity, the Pythagorean identity can be used for rewriting trigonometric expressions in equivalent, more useful, forms.
The Pythagorean theorem also allows us to convert between the sine and cosine values of an angle, without knowing the angle itself. Consider, for example, the angle theta in Quadrant start text, I, V, end text for which sine, left parenthesis, theta, right parenthesis, equals, minus, start fraction, 24, divided by, 25, end fraction. We can use the Pythagorean identity and sine, left parenthesis, theta, right parenthesis to solve for cosine, left parenthesis, theta, right parenthesis:
\begin{aligned} \sin^2(\theta)+\cos^2(\theta)&=1 \\\\ \left(-\dfrac{24}{25}\right)^2+\cos^2(\theta)&=1 \\\\ \cos^2(\theta)&=1-\left(-\dfrac{24}{25}\right)^2 \\\\ \sqrt{\cos^2(\theta)}&=\sqrt\dfrac{49}{625} \\\\ \cos(\theta)&=\pm\dfrac{7}{25} \end{aligned}
The sign of cosine, left parenthesis, theta, right parenthesis is determined by the quadrant. theta is in Quadrant start text, I, V, end text, so its cosine value must be positive. In conclusion, cosine, left parenthesis, theta, right parenthesis, equals, start fraction, 7, divided by, 25, end fraction.
Problem 1
theta, start subscript, 1, end subscript is located in Quadrant start text, I, I, I, end text, and cosine, left parenthesis, theta, start subscript, 1, end subscript, right parenthesis, equals, minus, start fraction, 3, divided by, 5, end fraction .
sine, left parenthesis, theta, start subscript, 1, end subscript, right parenthesis, equals

Want to try more problems like this? Check out this exercise.

## Want to join the conversation?

• How come these "Review" sections aren't in every subtopic? I think for those of us who don't find videos particularly effectively, having something we can read are really fantastic resources.
• There originally were no review sections and no articles. The site only had videos and practice exercises. Over the last year or so they have been adding in these review sections. They probably just haven't done all of them yet.
• What are real life ways to use this awesome proof? I MUST find out! :)
• There are countless real-life situations that use the Pythagorean identity. A great example is in architecture. If you're creating a blueprint of a structure that consists of right triangles and you would like to know the length of a side, the Pythagorean identity will help you do so. Geologists or explorers use it to find the height of a mountain with great accuracy. Not to mention how important it is in space when you can't always measure distances between objects easily. Here is a Prezi on many real-world applications of the trig identities (which was not made by me): https://prezi.com/vvsb1nqexnzd/trigonometric-identities-in-the-real-world/
• how do you get 16 from 3x3?
• this may not be helpful anymore, but they subtracted 9/25 from 1 and got 16/25.
• As far as I think the 3rd and 2nd quadrant is negative and the 4th and 1st quadrant is positive.
• For the cosine function, yes.
For the sine function, I and II are positive, III and IV are negative.
For the tangent function, I and III are positive, II and IV are negative.
• In the example above they calculated that 1-(−24/25)^2=sqrt(49/625) when taken sqrt of cos^2(θ). Could someone explain to me how did they get that solution?
• Hi Nenand,

Let me elaborate on this-

cosˆ2(θ) = 1 - (-24/25)ˆ2
cosˆ2(θ) = (625/625) - (576/625) (Do you remember that any number dived by itself is 1? Hence, 625/625 = 1)
cosˆ2(θ) = (625 - 576)/ 625
cosˆ2(θ) = 49/ 625
√cosˆ2(θ) = √49/ 625
cos(θ) = +-(7/ 25)

As the angle is in the IV quadrant, cos(θ) will be positive, i.e., (7/25).

I hope this helped.

Aiena.
• What does it mean by saying that "This identity is true for all real values of θ "?
• The Pythagorean Identity does not hold true when θ is a non-real number. If θ were an imaginary or complex number, for example, the identity might not be true.
• In one of my calculus problems it says that sin^2(-x) + cos^2(-x) = 1. Could someone please explain this? My textbook is less than helpful. :(
• sin(-x) = -sin(x)
cos(-x) = cos(x)

sin(-x)^2 + cos(-x)^2
= (-sin(x))^2 + cos(x)^2
= (-1)^2 * sin(x)^2 + cos(x)^2
= 1 * sin(x)^2 + cos(x)^2
= sin(x)^2 + cos(x)^2

sin(x)^2 + cos(x)^2 is just the Pythagorean identity so we know that it equals 1
• I don't understand how the definitions of sine, cosine, and tangent apply outside of a right triangle. If it's not a right triangle, then there are no more opposite sides or hypotenuses. How do they work when not being applied to a right triangle?
• The simple SOH CAH TOA definition of trig functions is not sufficient for angles greater than or equal to 90˚ (or lesser than or equal to 0˚). To evaluate the trig functions for other angles, we need to extend our definition of trig functions. This extension is accomplished by something called the "unit circle". Using this tool, we can evaluate the sine and cosine (and thus the tangent) of any angle. You can watch the videos on the unit circle in the Trigonometry playlist.
• How do you get 16 from 3x3 in the first problem? It say - (- 3/5) squared, and they get (16/25). That doesn’t even make sense!
• The right side of the equation in the explanation is 1 - (-3/5)^2, which is equal to 16/25:

sin^2(theta) = 1 - (-3/5)^2
= 1 - (-3/5) * (-3/5)
= 1 * (25/25) - (9/25)
= (25/25) - (9/25) = 16/25