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

Inquiry Activity Summary:

1. Where does sin^2x+cos^2x=1 come from?
A) First off, an identity is a proven fact or formula that is always true. The Pythagorean Theorem is an identity because it will always work, there is no number that cannot be plugged into that equation.
B) Using the letters x, y, and r we can create the Pythagorean Theorem. The Pythagorean Theorem comes from these letters. We can use x as one side, y as the other side, and r as the hypotenuse-remember the Pythagorean Theorem is for triangles- and you square all of them. Your result is the equation of x^2+y^2=r^2. These are terms we have derived from the unit circle and that is why we use them so that they can be associated with the trigonometric ratios.
C) Now we want the theorem equal to one so that we can create the identity. In order to do this we have to divide each letter by r^2. This leaves us with (x^2/r^2)+(y^2/r^2)=1. With the law of exponents we can make this a little bit easier to see and understand by making it a fraction squared for both. This leaves us with (x/r)^2+(y/r)^2=1. This is easier to understand and read.
D) Now we know that the ratio for cosine in the unit circle is (x/r). Remember that this comes from our triangles and the values they represent.
E) We also know that the ratio for sine is (y/r) which also comes from our triangles and the unit circle.
F) If you pay attention you will see that the ratios in d and e match the ratios we have in c. Using the substitution property we can replace our fractions with the functions and we have our new identity. We have cos^2(x)+sin^2(x)=1. This is what our identity is.
G) It is known as a Pythagorean Theorem Identity because all we used was the Pythagorean Theorem (which is an identity also) to figure it out and substitute some terms. It is the only formula that we used though.
H) We can verify this by using one of the angles from the unit circle. We will use a 60 degree angle. The cosine is (1/2)^2+ the sine which is (radical3/2)^2 and that produces the answer we are looking for which is one. Overall it is a working and true formula therefore it is an identity. You can also make it work with the 30 degree or 45 degree angles.

2. Show and explain how to derive the other two Pythagorean Theorem Identities from the first..
A) To get the second identity we need to divide everything in the equation by cos^2(x). This then gives us three fractions that fit with our other identities. The first is (cos^2x)/(cos^2x) which gives us the answer 1. The second is (sin^2x)/(cos^2x) which is equal to the powered up form of the reciprocal identity of tanx=sinx/cosx [Powered Up means that it is in the second or third or fourth power for each of the pieces of the identity]. That gives us tan^2x. Then the third, on the other side of the equal sign, is 1/cos^2x which is the powered up form of the reciprocal identity of secx=1/cosx. That gives us sec^2x. The resulting equation is 1+tan^2x=sec^2x. It is as easy as that.
B) To get the third one you divide the first identity by sin^2x. That gives you three fractions. The first is (cos^2x)/(sin^2x) which is the powered up form of the ratio identity of cotx=cosx/sinx. That gives us the answer cot^2x. The second fraction is sin^2x/sin^2x which is just 1. The third one on the other side of the equal sign is 1/sin^2x which is the powered up form of the reciprocal identity of cscx=1/sinx. That gives us the answer of csc^2x. The final equation-place the one in the front just to make it match the second one- is 1+cot^2x=csc^2x.

Inquiry Activity Reflection:

1) The connections that I see between Units N,O,P, and Q so far are that they are all focus on using the trigonometric ratios of sine, cosine, tangent, cosecant, secant, and cotangent. They also all are associated with the Unit Circle. Yeah for the Unit Circle!!

2) If I had to describe trigonometry in three words they would be ratios, triangles, and identities.