1. The Pythagorean identity, sin^2x+cos^2x=1, comes from the Pythagorean Theorem. If we arranged the Pythagorean theorem using x, y, and z, we would get x^2 + y^2 = r^2. And, if we made this equation equal one, we would have to divide everything by r^2. So, we would get: (x^2)/(r^2) + (y^2)/(r^2) = 1. We already know the trig ratio for cosine is x/r and the ratio for sine is y/r. So, (x^2)/(r^2)= cos^2 and (y^2)/(r^2) = sin^2. Therefore, for our final answer, we would get: cos^2 + sin^2 = 1, our Pythagorean theorem. Look at the pictures below for a visual. Also, we can prove this by plugging in values from the unit circle. The example below uses the coordinates of the 30 degree reference angle: (rad 3/2, 1/2).
2. To derive the two remaining Pythagorean Identities from sin^2x+cos^2x=1, divide it by cos^2x and sin^2x. When you divide it by cos^2x, you shuold get 1 + tan^2x = sec^2x. Then, when you divide it by sin^2x, you should get 1 + cot^2x = csc^2x. Look at the pictures below to see, step by step, how I got them.
INQUIRY ACTIVITY REFLECTION
1. "The connections that I see between Units N, O, P, and Q so far are:" how the Pythagorean identities are derived from the Pythagorean Theorem (x^2 + y^2 = r^2) -> ( x^2/r^2 + y^2/r^2 = 1) ->(cos^2x + sin^2 = 1) and how the reciprocal and ratio identities are derived from the trig ratios (cotx= cosx/sinx) -> (x/y) / (y/r) = x/y
2. "If I had to describe trigonometry in THREE words, they would be:" triangles, identities, and ratios.
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