I/D #3: Unit Q - Pythagorean Identities

INQUIRY ACTIVITY SUMMARY:

1.) Where does sin^2s+cos^2x=1 come from to begin with? You should be referring to Unit Circle ratios and the Pythagorean Theorem in your explanation.
       
       a. What is an "identity"? Why is the Pythagorean Theorem an "identity"?
          An identity is a proven facts and formulas that are true. The Pythagorean Theorem is an identity because it is proven to always true.

       b. Why is the Pythagorean Theorem using x, y, and r?
         The Pythagorean Theorem uses x, y, and r because based on a right triangle in quadrant 1. It uses x, y, and r are like a, b, and c for the Pythagorean Theorem.

         


       c. Perform an operation that makes the Pythagorean Theorem equal to 1.
           We would have to divide r^2. In order to make the formula equal to 1.




      d. What is the ratio for cosine on the unit circle?
          The ratio for cosine on the unit circle is x/r. Based on SOHCAHTOA, cosine is adjacent over  hypotenuse.

           


      e. What is the ratio for sine on the unit circle?
          The same as the top one above. The ratio for sine is opposite over hypotenuse.

        

      f. What do you notice about part (c) in relation to parts (d) and (a)? What can you conclude?

   

      g. What is sin^2x+cos^2x=1 referred to as a Pythagorean Identity?
           sin^2x+cos^2x=1 is referred as the Pythagorean Identity because in order to find it we use the Pythagorean Theorem.

      h. Choose one of the "Magic 3" ordered pairs from the Unit Circle to show that this identity is true.  

          



2.) Show and explain how to derive the two remaining Pythagorean Identities from sin^2x+cos^2x=1. Be sure to show step by step.
       a. Perform a single operation fairly to derive the identity with Secant and Tangent.

         


       b. Perform a single operation fairly to derive the identity with Cosecant and Cotangent.

         



INQUIRY ACTIVITY REFLECTION:

1.) "The connection that I see between Units N, O, P, and Q so far are..."that all of them refers back to the Unit Circle. Including the Pythagorean to find all the sides. As well as find the angles by using 180º and subtracting the number we have to find the missing angles.
2.) "If I had to describe trigonometry in THREE words, they would be..." interesting but difficult.