I/D #3: Unit Q Concept 1: Using Fundamental Identities to Simplify or Verify Expressions

Inquiry Activity Summary

Where does sin²x + cos²x  = 1 come from to begin with? 

First off, in order to understand the concept, we must acknowledge that an identity is a proven fact and formula that is always true. Therefore, the Pythagorean Theorem is an identity because it is a proven formula that is always true. The Pythagorean Theorem is usually demonstrated through the formula a^2+b^2=c^2. However, in reference to the unit circle, the Pythagorean Theorem is x^2+y^2=c^2. To get the theorem equal to 1, we divide both of the sides by r^2. The result is x^2/r^2+y^2/r^2=1. These can also be written as (x/r)^2+(y/r)^2=1. Now, it is in complete reference to the unit circle! Cosine for the unit circle is x/r and sine is y/r. Therefore, if you substitute the equation with cos^2θ and sin^2θ. The new equation, cos^2x+sin^x=1 is now referred as an Pythagorean Identity. We can prove this by plugging in a pair from the unit circle such as 60 degrees. The pair for 60 degrees is (1,2, radical 3 over 2). When plugging it into the formula, after simplifying, you get 1=1. IT IS TRUE. :D 

How to derive the two remaining Pythagorean Identities from sin2x+cos2x=1.

This shows how to get tan^2x+1=sec^2x. First, you have to divide both sides by cos^2x to get tangent and secant. cos^2x/cos^2x cancels out to become 1. After, using your identities, you simplify the equation. Thus, with our memory of identities, we get the identity tan^2x+1=sec^2x.

This Pythagorean Identity consists of cotangent and cosecant. To get there, we divide by sin^2x for both sides. We know that sin^2x/sin^2x cancels out to become 1. We then use the identity for cos^2x/sin^2x which is cot^2x. Then, we know that 1/sin^2x is csc^2x. The final answer 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 it uses the unit circle and trig functions. Also, it uses the Pythagorean Theorem as we have been using in triangles.
2. “If I had to describe trigonometry in THREE words, they would be crazy, mind-blowing, interesting.