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Pythagorean identity review

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

What is the Pythagorean identity?

This identity is true for all real values of θ. It is a result of applying the Pythagorean theorem on the right triangle that is formed in the unit circle for each θ.
A unit circle on an x y coordinate plane where the center of the unit circle is at the origin and the circumference of the circle touches (one, zero), (zero, one), (negative one, zero), and (zero, negative one). A point located at one thirty on the circle has the coordinates (x, y). A solid line slants up from left to right connecting from the origin to the point. The length of the solid line is measured one. A dashed line travels down vertically from the point to the x-axis around seven tenths. It has a length of sine of theta. The length from the origin to where the vertical dashed line is cosine of theta. The angle measure between the solid line and the x axis is 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 θ in Quadrant IV for which sin(θ)=2425. We can use the Pythagorean identity and sin(θ) to solve for cos(θ):
The sign of cos(θ) is determined by the quadrant. θ is in Quadrant IV, so its cosine value must be positive. In conclusion, cos(θ)=725.
Problem 1
θ1 is located in Quadrant III, and cos(θ1)=35 .

Express your answer exactly.

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

Want to join the conversation?

  • blobby green style avatar for user gameboyjustin
    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.
    (137 votes)
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  • orange juice squid orange style avatar for user Matthew Johnson
    What are real life ways to use this awesome proof? I MUST find out! :)
    (20 votes)
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    • piceratops ultimate style avatar for user Justin
      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/
      (57 votes)
  • blobby green style avatar for user james  jensen
    how do you get 16 from 3x3?
    (20 votes)
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  • starky ultimate style avatar for user .
    As far as I think the 3rd and 2nd quadrant is negative and the 4th and 1st quadrant is positive.
    (9 votes)
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  • leaf grey style avatar for user philosopher
    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?
    (11 votes)
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    • blobby green style avatar for user Aiena
      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.

      (25 votes)
  • stelly blue style avatar for user ks
    Im still confused with the quadrants things. Can someone list all the positive/negative values for the quadrants? Thanks!
    (7 votes)
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    • blobby green style avatar for user Lucas Salazar
      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!
      (30 votes)
  • piceratops ultimate style avatar for user 𝓜𝓪𝓱𝓮𝓼𝓱 𝓜𝓪𝓱𝓮𝓷𝓭𝓻𝓪𝓴𝓪𝓻
    What does it mean by saying that "This identity is true for all real values of θ "?
    (11 votes)
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  • female robot ada style avatar for user Chloe
    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. :(
    (9 votes)
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  • starky sapling style avatar for user 20leunge
    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?
    (6 votes)
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    • mr pink red style avatar for user andrewp18
      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.
      (18 votes)
  • winston default style avatar for user Daksh Naidu
    nothing makes sense
    (10 votes)
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