Pythagorean Theorem:
To start the “Measuring Your World” project we had to prove why the Pythagorean theorem worked. To do this we needed to know the equation of a^2+b^2=c^2. We started off with fairly easy assingment. Each week we would have a different packet that was due and each week it got harder as we progressed with knowledge of the content being taught. By the end of the month we had become semi experts in trigonametry.
We can then use this equation to derive the equation of a circle centered at the origin of a Cartesian coordinate plane, also known as a unit circle. This circle has a radius of 1, which makes it a begining point. All points of any circle, including the unit circle are equally distant from a given point, these points are known as locus. Since we have proved all circles are similar because they are regular and if overlapped on its origin, can be dilated to prove that it's simlar. This equation is x^2 + y^2 = r^2, which follows a similar form to both the pythagorean theorem and the distance formula. To find points on the unit circle, at 30 degrees, 45 degrees and 60 degrees we followed the same set of steps. We use the formula x^2 + y^2 = 1
When solving for 45 degrees, from the information we are given, we can conclude the x and y are equal and proceed with solving.
For solving for 30 degrees, we can reflect the triangle over the x-axis creating an equilateral triangle and solving with the knowledge that the side length of y is now ½ since being reflected. This gives us enough information to finish solving alike to the problem previously.
Last but certainly not least, if we are solving for a triangle at 60 degrees, we can solve this very similarly to how we solved the 30 degree triangle, by reflecting over the y-axis and solving.
We thankfully learned about each of these trigonomatry formulas by using a shape as simple as a right triangle. To start off, labeling the triangle to where the delta angle is located helps to show which lengths you will working with, the sides are labeled: opposite, adjacent, and hypotenuse. We learned about sine by knowing that the equation S=O/H (or sine equals opposite divided by adjacent) If we are looking for a y-coordinate on the unit circle we can use sine to solve. If we are instead looking for the intersection of a point on the x-axis on the unit circle, we can use cosine to solve. The equation used to solve for cosine is written as C=A/H.
We were able to find the answer by finding the distance between to points then finding the angle of those lines to find the missing angle to find the missing side length. Sin(A)/a = Sin(B)/b = Sin(C)/c which is the law of sines. Along with the Law of Sines we were able to derive the Law of cosines, this function helps us find the missing angles on a non-right triangle, when you are only given two side lengths and one angle. The formula was (c^2=a^2+b^2 - 2abcosϑ).
8 Trigonometric Formulas:
The next set of terms we covered are ArcSine, ArcCosine, ArcTangent. These can each be found by finding the inverse of either sine, cosine, or tangent. Since we know that the inverse of an object will always be a negative so we learned this concept by understanding that ArcCosine is: angle theta=cos^-1 and so on for sine and tangent. We also learned about the Law of Sine. An assingment when we used this formula was the Mt. Everast assingment, in this worksheet we had to find the distance from one point in india to the Himalaya Mt. Everast. We had to find how the english found it, by finding two distance points and then plugging in the angles to find the missing distance. Written out the law of sines is sinB/b = sinC/c = sinA/a. The law of cosines is used when we know two lengths and one angle between them, this is written as c2 = a2 + b2 – 2ab cos C. Each of these formulas can be applied to any problem with or without a right triangle that requires looking for a length or height.
Slide Show Below:
To start the “Measuring Your World” project we had to prove why the Pythagorean theorem worked. To do this we needed to know the equation of a^2+b^2=c^2. We started off with fairly easy assingment. Each week we would have a different packet that was due and each week it got harder as we progressed with knowledge of the content being taught. By the end of the month we had become semi experts in trigonametry.
We can then use this equation to derive the equation of a circle centered at the origin of a Cartesian coordinate plane, also known as a unit circle. This circle has a radius of 1, which makes it a begining point. All points of any circle, including the unit circle are equally distant from a given point, these points are known as locus. Since we have proved all circles are similar because they are regular and if overlapped on its origin, can be dilated to prove that it's simlar. This equation is x^2 + y^2 = r^2, which follows a similar form to both the pythagorean theorem and the distance formula. To find points on the unit circle, at 30 degrees, 45 degrees and 60 degrees we followed the same set of steps. We use the formula x^2 + y^2 = 1
When solving for 45 degrees, from the information we are given, we can conclude the x and y are equal and proceed with solving.
For solving for 30 degrees, we can reflect the triangle over the x-axis creating an equilateral triangle and solving with the knowledge that the side length of y is now ½ since being reflected. This gives us enough information to finish solving alike to the problem previously.
Last but certainly not least, if we are solving for a triangle at 60 degrees, we can solve this very similarly to how we solved the 30 degree triangle, by reflecting over the y-axis and solving.
We thankfully learned about each of these trigonomatry formulas by using a shape as simple as a right triangle. To start off, labeling the triangle to where the delta angle is located helps to show which lengths you will working with, the sides are labeled: opposite, adjacent, and hypotenuse. We learned about sine by knowing that the equation S=O/H (or sine equals opposite divided by adjacent) If we are looking for a y-coordinate on the unit circle we can use sine to solve. If we are instead looking for the intersection of a point on the x-axis on the unit circle, we can use cosine to solve. The equation used to solve for cosine is written as C=A/H.
We were able to find the answer by finding the distance between to points then finding the angle of those lines to find the missing angle to find the missing side length. Sin(A)/a = Sin(B)/b = Sin(C)/c which is the law of sines. Along with the Law of Sines we were able to derive the Law of cosines, this function helps us find the missing angles on a non-right triangle, when you are only given two side lengths and one angle. The formula was (c^2=a^2+b^2 - 2abcosϑ).
8 Trigonometric Formulas:
The next set of terms we covered are ArcSine, ArcCosine, ArcTangent. These can each be found by finding the inverse of either sine, cosine, or tangent. Since we know that the inverse of an object will always be a negative so we learned this concept by understanding that ArcCosine is: angle theta=cos^-1 and so on for sine and tangent. We also learned about the Law of Sine. An assingment when we used this formula was the Mt. Everast assingment, in this worksheet we had to find the distance from one point in india to the Himalaya Mt. Everast. We had to find how the english found it, by finding two distance points and then plugging in the angles to find the missing distance. Written out the law of sines is sinB/b = sinC/c = sinA/a. The law of cosines is used when we know two lengths and one angle between them, this is written as c2 = a2 + b2 – 2ab cos C. Each of these formulas can be applied to any problem with or without a right triangle that requires looking for a length or height.
Slide Show Below: