Measuring Your World Reflection:
We began this project by studying the formula (Pythagoras Theorem), a^2 + b^2 = c^2. In the formula, c represents the hypotenuse, longest side. We used this to find a missing length of any right triangle, by plugging in known side lengths and factors. There is a vast amount of ways to prove the Pythagorean Theorem. We solved or worked with the formula by actually dividing a right triangle up and simply breaking it down into two smaller triangles. We are capable to make this proof by, taking apart the triangle, into 3 even parts and prove that a^2 + b^2 = c^2 are still correct. Next to make a proof of the distance formula, we can use a^2 + b^2 = c^2 .
When Using the Distance Formula to derive the equation of a circle centered at the origin of a Cartesian coordinate plane the pythagorean theorem tells you from all the point that lie at a equal distance from a given point. We have proved all circles are similar because they are regular. This means they can be overlapped on their origin and dilated to be the same size. The equation x² + y² = r² which follows a similar form to both the pythagorean theorem and the distance formula can be used to find points on the unit circle.The unit circle is defined as the circle whose radius is one. The sine is defined as the x-component of our angle, divided by the radius. The unit circle can be used to define things greater than one, because some trig functions attain values greater than 1. Finding points on the unit circle at a specific degree look at the terminal sides. For each angle that is drawn in a standard position there is a related angle known as the reference angle. That is the angle that will always be formed by the terminal side and x-axis.
The first step to finding the point on the unit circle is drawing a line straight down from the y-coordinate to the x-coordinate. This is because it’ll create a right triangle making it easier for us to find the other sides of the triangle. Now because we made it a right triangle we now know that one of the angles is 90 degrees, and because we are solving for the 30 degree unit circle we also know that the other angle is 30 degrees leaving the last angle to be 60 degrees. Now to make it easier we reflect the triangle over the line, because we know that the far side angle is 60 degrees we know the reflected angle is 60 degrees which changes that 30 degree angle to also equal 60 degrees. This makes it an equilateral triangle, and because we know one side equals one then the other two sides also equal one. Because we reflected the triangle across the line we know that the line we drew in the beginning now equals ½. Now all you do is plug in for the equation x^2+y^2=1 so ½^2+y^2=1, y = ¾ , ¼+¾=1, now we simplify it to the (square root of 3 over 2), (one half).
Using the symmetry of a circle to find the remaining points on the unit circle You can plug values in to your equation. The values can use symmetry to find the rest of the points on the unit circle. We can now use symmetry to find the rest of the points on the unit circle. Now all we have to do is reflect the points off the x axis then the y axis. Then get a complete unit circle. For example if one of our points is (square root of 3 over 2, one half) then the reflected point off of the x axis would be (square root of 3 over 2, negative one half)
Using the unit circle to define sine and cosine (of the angle theta) Everything that make up a unit circle, is reflecting the points off of x/y-axis. Sine is "The length of the opposite side divided by the length of the hypotenuse", sine also corresponds with the y-coordinate for the point on the unit circle. When a radical line makes a angle of theta degrees with the x- axis.
Defining the tangent function:
In a right triangle the tangent of the angle is the opposite side of the triangle divided by the adjacent side of the triangle. Tanø=opp/adj a way to remember this is TOA. The length of the opposite side is divided by the length of the adjacent side. tan θ = opposite / adjacent. But the ratio would be make of y/x. and Tangent is equal to sin of a angle over cosine of a angle.
Using the Mount Everest problem to discover the Law of Sines ("taking apart") In class we attempted the Mount Everest problem. Which related real world things and we explained the math behind it. In the image above it shows a triangle with sides labeled ABC. I knew two of the three side lengths. So were AB was it was 170 and angle A was 72 degrees and angle B was 49 degrees. So with knowing the other two angles we need to subtract each angle by 180. The number that you receive is the last angle because a triangle has to add up to 180. So angle C is 59 degrees. The first step was to go half way between points B and C. And that new point will be called point D. Next we use the sine fiction to figure out the height of the triangle. The number that I received was 128.30. Using those two side lengths and the pythagorean theorem I found out that the side lengths of D and B is 111.53. Then I just repeated those steps for the other side and found that AC was 149.68 and CD was 77.09.
Deriving the Law of Sines
The first step that you will have to take is labeling the triangle, and trying to find the missing angles with the information that you already have. By plugging in what you know into this equation a/sinA=b/sinB. Next you find the second missing side by using the next equation a/sinA=c/sinC. So the Law of Sines would be a/sin(a)=b/sin(b)=c/sin(c).
Deriving the Law of Cosines The law of Cosines is pretty straightforward forward. It is c^2=a^2+b^2-2abcos. If you have side lengths A and B, and you are needing to find side length C, all you need to do is plug in this equation.
Measuring Your World: Part 2
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