Inquiry Activity Summary
The Pythagorean is here again. We first saw it in unit N in the unit circle, and it relates to trig identities as well. Using x, y, and r, we can show that. (x^2/r^2)+(y^2/r^2)=(r^2/r^2) which is also (x^2/r^2)+(y^2/r^2)=1. This can be simplified to (x/r)^2+(y/r)^2=1. sin2x+cos2x=1 comes from the unit circle. The ratio for cosine on the unit circle is x/r. The ratio for since is y/r. Here we are going to take one of the magic 3 ordered pairs: 30 degrees. cos30 is equal to radical 3 over 2 and sin30 is equal to 1/2. Radical 3 over 2 squared is equal to 3/4. 1/2 squared is equal to 1/4. 3/4+1/4=1 which is where you get the one on the right side. Since we squared these fractions, we have to show it in our identities also. This identity is actually the Pythagorean theorem moved around, hence it is also called the Pythagorean identity. Keep in mind that an identity is "a proven fact and formula that is always true." This tactic works with any degree from the magic 3.
We can also derive the secant and tangent from sin2x+cos2x=1. If you divide all of that by cos2x, you will end up with tan2x+1=sec2x. With tanx=sinx/cosx, multiply the left and right sides by themselves to get a squared value for both. tanx*tanx=(sinx/cosx)(sinx/cos). In the end you will get tan2x=sin2x/cos2x.
Inquiry Activity Reflection
- The connections that I see between Units N, O, P, and Q so far are that they all relate to triangle and they all use the six trig functions.
- If I had to describe trigonometry in THREE words, they would be challenging, connected, and unexceptional.
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