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 this is a Pythagorean identity. An identity is a proven fact or formula that is always true. The first thing we did was write out the Pythagorean theorem as show below. Then we replaced a^2+b^2=C^2 with x^2+y^2=r^2 ( as shown above) We are able to replace it with these variables because it consists of the same characteristic as the unit circle, were using all the variables from the first quadrant. At this moment our formula  is  x^2 + y^2 = r^2 and we want to get to sin2x+cos2x=1. To do that, we divide everything by r^2. If we divide everything by r^2, we get this: x^2/r^2 + y^2/r^2 = 1 (r^2/r^2).  So r^2/r^2 becomes 1 and we can put (x/r) ^2+ (y/r)^2=1. Then we should notice that these variables are the ratios of  cosine and sine. Now that we know this we can replace then and get a final answer of cos^2x+sin^2x=1.


Does it really work?

 

           In order to verify that it works we can plug in the 30,45 and 60 degree angles. ( All values from the unit circle) In this case I choose 60*. The 1/2^2 becomes 1/4 and the rad3/2 becomes 3/4 because the squared cancels the 3 and you just multiply 2 by 2. This gives us an answer of 1.

 

2. The other Pythagorean Identities.

 

 

 

 

 The picture above shows the steps of how to get tan^2x + 1 = sec^2x. First, you will have to divide both sides by cos^2x. For this one, we want tangent and secant. We can look at that as y/r times r/x. So as a result we have y/x which is tangent. Cosine divided by cosine is simply 1. And 1 divided by cos2x  which is sec2x because sec x = 1/cos x, we just powered up because everything is being multiplied in this reciprocal identity. (look at the picture above)

 

The other Pythagorean identity is 1+cot^2x=csc^2x. In this identity we have cot and csc.  We divide everything by sin2x . We divide cos^2x by sin^2x and we get cot^2x. We divide 1 by sin^2x and we end up with csc2x. This is how we get the answer.

 

INQUIRY ACTIVITY REFLECTION 

1.THE CONNECTIONS I SEE WITH UNIT N, O, P AND Q SO FAR ARE the trig functions and the ratios. Also the magic three from the unit circle to derive the things above.

2. IF I HAD TO DESCRIBE TRIGONOMETRY IN THREE WORDS, THEY WOULD BE sin, cos, and tan.