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.

how do you get 16 from 3x3?

this may not be helpful anymore, but they subtracted 9/25 from 1 and got 16/25.

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/

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.

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.

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*

Main content

Algebra 2

Course: Algebra 2 > Unit 11

Lesson 3: The Pythagorean identity

Pythagorean identity review

CCSS.Math:

HSF.TF.C.8

HSF.TF.C

Google Classroom

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.

Want to learn more about the Pythagorean identity? Check out this video.

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

Express your answer exactly.

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

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gameboyjustin
Posted 6 years ago. Direct link to gameboyjustin's post “How come these "Review" s...”
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.
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(108 votes)
Answer
- Kim Seidel
  Posted 6 years ago. Direct link to Kim Seidel's post “There originally were no ...”
  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.
  Button navigates to signup page
  (62 votes)
Matthew Johnson
Posted 6 years ago. Direct link to Matthew Johnson's post “What are real life ways t...”
What are real life ways to use this awesome proof? I MUST find out! :)
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(15 votes)
Answer
- Justin
  Posted 6 years ago. Direct link to Justin's post “There are countless real-...”
  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/
  Comment on Justin's post “There are countless real-...”
  (43 votes)
james jensen
Posted 3 years ago. Direct link to james jensen's post “how do you get 16 from 3x...”
how do you get 16 from 3x3?
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(16 votes)
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- Koryn Shaw
  Posted 3 years ago. Direct link to Koryn Shaw's post “this may not be helpful a...”
  this may not be helpful anymore, but they subtracted 9/25 from 1 and got 16/25.
  Comment on Koryn Shaw's post “this may not be helpful a...”
  (23 votes)
.
Posted 5 years ago. Direct link to .'s post “As far as I think the 3rd...”
As far as I think the 3rd and 2nd quadrant is negative and the 4th and 1st quadrant is positive.
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(5 votes)
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- Mark Geary
  Posted 5 years ago. Direct link to Mark Geary's post “For the cosine function, ...”
  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.
  Comment on Mark Geary's post “For the cosine function, ...”
  (29 votes)
philosopher
Posted 3 years ago. Direct link to philosopher's post “In the example above they...”
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?
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(5 votes)
Answer
- Aiena
  Posted 3 years ago. Direct link to Aiena's post “Hi Nenand, Let me elabor...”
  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.
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  (19 votes)
𝓜𝓪𝓱𝓮𝓼𝓱 𝓜𝓪𝓱𝓮𝓷𝓭𝓻𝓪𝓴𝓪𝓻
Posted 3 years ago. Direct link to 𝓜𝓪𝓱𝓮𝓼𝓱 𝓜𝓪𝓱𝓮𝓷𝓭𝓻𝓪𝓴𝓪𝓻's post “What does it mean by sayi...”
What does it mean by saying that "This identity is true for all real values of θ "?
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(7 votes)
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- KLaudano
  Posted 3 years ago. Direct link to KLaudano's post “The Pythagorean Identity ...”
  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.
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  (11 votes)
Chloe
Posted 3 years ago. Direct link to Chloe's post “In one of my calculus pro...”
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. :(
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(5 votes)
Answer
- KLaudano
  Posted 3 years ago. Direct link to KLaudano's post “sin(-x) = -sin(x) cos(-x)...”
  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
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  (9 votes)
20leunge
Posted 6 years ago. Direct link to 20leunge's post “I don't understand how th...”
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?
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(3 votes)
Answer
- andrewp18
  Posted 6 years ago. Direct link to andrewp18's post “The simple SOH CAH TOA de...”
  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.
  Button navigates to signup page
  (12 votes)
Roropie185
Posted 3 years ago. Direct link to Roropie185's post “How do you get 16 from 3x...”
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!
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(3 votes)
Answer
- Hayden
  Posted 3 years ago. Direct link to Hayden's post “The right side of the equ...”
  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
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  (8 votes)
msaifahsan
Posted 4 days ago. Direct link to msaifahsan's post “Anyone struggling with sa...”
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.
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