The identity sin2(x) + cos2(x) = 1 is the x only for right triangles?

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nick-teal's profile pic

nick-teal | High School Teacher | (Level 3) Adjunct Educator

Posted on

Every triangle we make in the unit circle will be a right triangle.  But x is not the right angle.

It will look something like the picture below.

*note how this will work for any angle theta*

So we use the pythagorean theorem where sin(theta)=y and cos(theta)=x to make:

`sin^2(theta) + cos^2(theta) = 1`

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steveschoen's profile pic

steveschoen | College Teacher | (Level 1) Associate Educator

Posted on

This formula is always true, for all angles.  So, not necessarily right triangles.  But, for all angles.  Such as:

`sin^2` (60) + `cos^2` (60) = 1

Both x's have to be the same angle.  But, it would be true for all angles.

Now, this identity is used a lot to describe the unit circle on a coordinate plane, Which can follow right triangles.  But, this identity is good for any angles.  The x's just have to be the same number.

kksxxx's profile pic

kksxxx | eNotes Newbie

Posted on

I asked the question.  Sorry, that second answer by me was posted in haste and has a typo.

I meant "therefore by definition, it is only for right triangles" (not right "angles"), because sine or cosine are only defined for right triangles, and not for any triangle. 

kksxxx's profile pic

kksxxx | eNotes Newbie

Posted on

I asked the question.  However, I believe the first answer is in error as follows. 

sine is the ratio of the length of the opposite side to the length of the hypotenuse. In our case, it does not depend on the size of the particular right triangle chosen, as long as it contains the angle A, since all such triangles are similar.  Therefore by definition, it is only for right angles. 

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