I/D 3 Unit Q- Pythagorean Identities

1. sin^2x and cos^2x=1 comes from the Pythagorean theorem of x^2+y^2=r^2. We know that x/r equals cosine and y/r equals sine from the unit circle ratios as we should know that our trigonometric functions of cosine is adjacent over hypotenuse and sine is opposite over hypotenuse. Since in the unit circle we considered a side opposite of an angle to be opposite of the angle that is protruding from (0,0) in all cases, we can deduce that the adjacent value equals x and that the opposite value equals y. If we take the Pythagorean theorem and solved to make r^2 1 by dividing the whole formula by r^2, we get what we wanted which is (x/r)^2+(y/r)^2=1. As we already went over, x/r is cosine and y/r is sine, and since the equation is already squared, that is where we get our equation of sin^2x and cos^2x=1. The x value that accompanies the formula can be substituted for theta if we never knew it.

2. In order to derive the identity with Secant and Tangent, we start off with our standard formula of sin^2x+cos^2x=1. We want to get secant, so we divide the whole equation by cos^2x in order to get that one the right side. Once we have 1/cos^2x on the right side, we see that same ratio is secant itself, just used twice, so we substitute 1/cos^2x for secx. On the left side of the equation, we also see sin/cos being the identity for tan, so we substitute tan^2x for that since sin/cos was squared as well. Cos^2x/cos^2x is 1 because they are the same number, just one over the other. We then obtain our final formula of tan^2x+1=sec^2x.

3. Just like the last problem, we start off with our formula of sin^2x and cos^2x=1. We want to get cosecane and cotangent this time, so we divide the equation by sin^2x this time. We get sin^2x/sin^2x+cos^2x/sin^2=1/sin^2x. For the right side, we can substitute that for csc^2x because we know that one of our trig identities is that csc equals 1 over sin. We can sub sin^2x/sin^2x for 1 since they equal each other. We substitute cos^2x/sin^2x for cotangent because the identity of cotangent is cos/sin, or x/y. After we substitute everything, we get 1+cot^2x=csc^2x as our final equation.

Inquiry Activity Reflection

1. THE CONNECTION I SEE BETWEEN UNITS N, O,P, AND Q SO FAR is that they all utilize the unit circle as well as the Pythagorean theorem to solve a multitude of problems. 

2. IF I HAD TO DESCRIBE TRIGONOMETRY IN THREE WORDS, THEY WOULD BE sine, cosine, and tangent.