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Trigonometric Identities - Concept
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
Identities are equations true for any value of the variable. Since a right triangle drawn in the unit circle has a hypotenuse of length 1, we define the trigonometric identies x=cos(theta) and y=sin(theta). In the same triangle, tan(theta)=x/y, so substituting we get tan(theta)=sin(theta)/cos(theta), the tangent identity. Another key trigonometric identity sin2(theta) + cos2(theta)=1 comes from using the unit circle and the Pythagorean Theorem.
I want to talk about trigonometric identities. Now recall an identity is an equation that is true for all applicable values of the variable. Here are 2 examples. x squared minus 1, the difference of squares is x plu- x-1 times x+1 and ln of e to the x equals x. I want to find some trigonometric identities and the unit circle can help me out.
So I've got the unit circle drawn over here. Remember the unit circle is the circle with equation with equation x squared plus y squared equals 1. So it's got radius 1 center to the origin and that the sine and cosine are defined in the following way. x equals the cosine of theta and y equals the sine of theta. And remember the tangent of theta is y over x, and therefore tangent equals sine over cosine. Sine over cosine.
Note also that combining these these 2 sets of information, we get the Pythagorean identity, cosine squared theta plus sine squared theta. So these are going to be really important identities and this idea of using the unit circle to derive new identities is what we're going to use in upcoming episodes.
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