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

INQUIRY SUMMARY ACTIVITY

1. Where does sin^2x + cos^2x=1 come from to begin with?

     An "identity" is proven facts and formulas that are always true. The Pythagorean Theorem is an identity because it has been proven to be always true. Since the Pythagorean Theorem consists of x, y, and r, it is also the same as a, b, and c.

    Since a^2 + b^2=1, we divide r^2 on both sides. After MEMORIZING the Unit Circle, we knew cosine is (x/r)^2 and (y/r)^2 is sine. The ratio for cosine on the unit circle is x/r and the ratio for sine is y/r.

Sin^2x + cos^2x=1 is referred to as a Pythagorean Identity because we used the Pythagorean Theorem to find it.

  I will choose one of the “Magic 3” ordered pairs (30, 45, or 60 degrees) from the Unit Circle to show that this identity is true; in this case, I chose 60 degrees.

2). Show and explain how to derive the two remaining Pythagorean Identities from sin^2x + cos^2x=1.

The picture below shows and explains how to get tan^2x + 1 = sec^2x. The identity with Secant and Tagent is tan^2x + 1 = sec^2x.

The picture below shows and explains how to get 1 + cot^2x = csc^2x. The identity for Cosecant and Cotangent 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 the Unit Circle and its ratios play a big part in all these units so “MEMORIZE THEM” and they also involve triangles to use the Pythagorean Theorem to get the answer or the missing side/angle.

2). If I had to describe trigonometry in THREE words, they would be burdensome, exhausting, and painful.