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## Algebra II (2018 edition)

### Course: Algebra II (2018 edition)>Unit 9

Lesson 6: The Pythagorean identity

# Pythagorean identity review

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

## What is the Pythagorean identity?

${\mathrm{sin}}^{2}\left(\theta \right)+{\mathrm{cos}}^{2}\left(\theta \right)=1$
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 $\text{IV}$ for which $\mathrm{sin}\left(\theta \right)=-\frac{24}{25}$. We can use the Pythagorean identity and $\mathrm{sin}\left(\theta \right)$ to solve for $\mathrm{cos}\left(\theta \right)$:
$\begin{array}{rl}{\mathrm{sin}}^{2}\left(\theta \right)+{\mathrm{cos}}^{2}\left(\theta \right)& =1\\ \\ {\left(-\frac{24}{25}\right)}^{2}+{\mathrm{cos}}^{2}\left(\theta \right)& =1\\ \\ {\mathrm{cos}}^{2}\left(\theta \right)& =1-{\left(-\frac{24}{25}\right)}^{2}\\ \\ \sqrt{{\mathrm{cos}}^{2}\left(\theta \right)}& =\sqrt{\frac{49}{625}}\\ \\ \mathrm{cos}\left(\theta \right)& =±\frac{7}{25}\end{array}$
The sign of $\mathrm{cos}\left(\theta \right)$ is determined by the quadrant. $\theta$ is in Quadrant $\text{IV}$, so its cosine value must be positive. In conclusion, $\mathrm{cos}\left(\theta \right)=\frac{7}{25}$.
Problem 1
${\theta }_{1}$ is located in Quadrant $\text{III}$, and $\mathrm{cos}\left({\theta }_{1}\right)=-\frac{3}{5}$ .
$\mathrm{sin}\left({\theta }_{1}\right)=$

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.
• Im still confused with the quadrants things. Can someone list all the positive/negative values for the quadrants? Thanks!
• This acronym helped me a lot: ASTC

A / ALL: This is the first quadrant (I), all values are positive (Sine, Cosine, Tangent, Cosecant, Secant and Cotangent)

S / Sine: This is the second quadrant (II), only the Sine and its inverse Cosecant (Wich you will see further in this course) are positive. Cosine, Tanget and its inverses are negative.

T / Tanget: This is the third quadrant (III), only the Tanget and its inverse Cotangent are positive. Cosine, Sine and its inverses are negative.

Cosine / Cosine: This is the fourth (IV), only the Cosine and its inverse Secant are positive. Sine, Tangent and its inverses are negative.

Summing up:

"S" Second quadrant (II) | "A" First quadrant (I)    Sine is positive      |    All are positive  ________________________|________________________                         |"T" Third quadrant (III) | "C" Fourth quadrant (IV)   Tangent is positive   |    Cosine is positive

Hope it helps!
• 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.
• Here's what you would put in a calculator so that you don't have to write everything out.

The angle (θ) is located in Quadrant IV and cos(θ) = 3/5

(3/5)^2=9/25
1-9/25=16/25
sqrt16/25= ±4/5

Now you just have to determine if it is positive or negative depending on which quadrant you're solving for and if you're looking for the x- or y-coordinate.

For this particular problem, the answer will be -4/5, because in Quadrant IV, you have (+cos(θ), -sin(θ))

Hopefully, this helps :)