I/D# 3: Unit Q Concept 1: Pythagorean Identities

Pythagorean Identities

INQUIRY ACTIVITY SUMMARY:

Fig 1 (http://www.regentsprep.org/Regents/math/algtrig/ATT9/pythagoreanid.htm)

1. Where does sin2x+cos2x=1 come from?

a) An "identity" by definition according to Mrs. Kirch, is a "proven fact or formula that is always true". The Pythagorean theorem is an identity because it will work with any triangle that is legitimately a right triangle. Because right triangles reoccur in the unit circle (in 30, 60, and 45 degree angle smaller triangles), the Pythagorean theorem can be found all throughout.

b) The Pythagorean theorem is side x squared plus side y squared equals the hypotenuse (c) squared. These terms only work when we have a right angle, and luckily, we do in all 3 types of our unit circle triangles (30, 60, and 90).

Fig 2 (http://image.tutorvista.com/cms/images/38/pythagorus-theorem.JPG)

c) We can make the Pythagorean theorem equal to 1 simply by dividing both dies by c^2. This can be then simplified by writing the squared outside parenthesis, signifying everything inside is squared. What's left inside the parenthesis should look very familiar :)

What's left after simplifying is (x/r)^2+(y/r)^2=1

d) The ratio for cosine on the unit circle is (x/r),with x being "adjacent" and r being "radius". In the unit circle, r is always 1 so cosine can also be represented as simply "x" (x/1).

e) The ratio for sine on the unit circle is (y/r), with y being "opposite" and r being "radius". In the unit circle, the radius is always equal to 1 so sine can also be represented as simply "y" (y/1). 

f) The inside of the parenthesis in part c can be represented by the ratios we found.  (x/r)^2+(y/r)^2=1 becomes (cosine)^2+(sine)^2=1. I can conclude that the Pythagorean theorem is always true because it is a THEOREM, therefore the relationship I just came up with after simplifying is also true.

Fig 3(http://www.regentsprep.org/Regents/math/algtrig/ATT9/pythagoreanid.htm)

g) sin2x+cos2x=1 is referred to as a Pythagorean identity because when a circle is formed out of the unit circle with a radius of 1, the legs of that triangle can have ratios that are over 1 (cosine and sine) and thus represented simply as x and y. The picture above depicts this sort of triangle in the unit circle Because it is a right triangle, x^2 (leg 1 and also cosinex) + y^2 (leg 2 and also sinex) = 1 (radius in unit circle). 

h)Our "magic pair" of a 30 degree (rad3/2, 1/2) triangle can prove the validity of the Pythagorean identity.

2. You can derive the two remaining Pythagorean Identities from sin2x+cos2x=1 easily with both secant and tangent (a) and cosecant and cotangent (b).

a)

 Fig 4 (http://www.regentsprep.org/Regents/math/algtrig/ATT9/pythagoreanid.htm)

 Using our fundamental Pythagorean identity of sin2x+cos2x=1 we can derive a second equation that contains tangent and secant. Simply divide the formula by cos^2x. According to our ratio identities, sinx/cosx=tanx. We also know that the inverse of cosine (1/cosx) is secantx. By plugging in these new values we get tan^2x+1=sec^2x.

b)

Fig 5 (http://www.regentsprep.org/Regents/math/algtrig/ATT9/pythagoreanid.htm)

Starting off we our fundamental pythegorean identity of sin2x+cos2x=1, we can divide by sinx^2x. We know cosx/sinx=cotangentx because it is a ratio identity. We also know the inverse of sine (1/sinx) is cosecant x. Therefore, if we plug in those new values we know 1+cot^2x=csc^2x.

INQUIRY ACTIVITY REFLECTION:

1. The connections that I see between Units N, O, P, and Q so far are... the same trigonometric ratios, the same validity tests that can be proven with our magic pair triangle coordinates, and the repeated use of the unit circle (with radius of 1 of course).

2. If I had to describe trigonometry in THREE words, they would be... ratios, identities, and derivations.

REFERENCES:

Fig 2: http://image.tutorvista.com/cms/images/38/pythagorus-theorem.JPG

Fig 1,3,4,5: http://www.regentsprep.org/Regents/math/algtrig/ATT9/pythagoreanid.htm