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## Algebra 2

### Course: Algebra 2>Unit 11

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
• Current
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.
• Anyone struggling with say whether a cos or sin of an angle is positive or negative. Remember, ASTC which means the first quadrant's all Sine, Cosine, and Tangent are positive which is why A stands for "All". The second quadrant's only Sine is positive while the Cos and Tan are negative which is why S stands for sine. The third is only Tan that's positive because T is for "Tan" and the fourth one is only Cos that's positive because C is for "Cos". Remember this by "All Students Take Calculus" as someone taught me. • 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? • 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.
• 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!   • 