Some Triangle DefinitionsSine : The trigonometric function that is equal to the ratio of the side opposite a given angle (in a right triangle) to the hypotenuse.
Cosine : The trigonometric function that is equal to the ratio of the side adjacent to an acute angle (in a right-angled triangle) to the hypotenuse. Tangent: A line which touches a circle or ellipse at just one point. Below, the blue line is a tangent to the circle. The radius to the point of tangency is always perpendicular to the tangent line. ArcSine : The arcsin function is the inverse of the sine function. It returns the angle whose sine is a given number. |
ArcCosine : Inverse Cosine - the Inverse function of the Cosine; the angle that has a cosine equal to a given number.
ArcTan : The arc tan function is the inverse of the tangent function. It returns the angle whose tangent is a given number.
The Law of Sines: The law of sines is the relationship between the sides and angles of non-right triangles . Simply, it states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all sides and angles in a given triangle.
Law of CoSines: A law stating that the square of a side of a plane triangle is equal to the sum of the squares of the other two sides minus twice the product of the other sides multiplied by the cosine of the angle between them.
Narratives:
When it comes proving the Pythagorean Theorem in solving problems that relate to 2-dimensional shapes the Pythagorean Theorem sums up the areas of two small squares that equal the away of the bigger one. This leads to the pythagorean theorem serving a basis of the distance formula. When using the pythagorean Theorem to derive the distance forumula it helps to start out by placing the right triangle in the coordinate plane and by giving it vertices of x and y. The Pythagorean Theorem states that the sum of the squares of the length of the legs is equal to the square of the hypotenuse. To be able to find the length of the "legs" (coordinates) we use absolute value. By using absolute value we are squaring the differences in the coordinates and then last by taking the square root of both sides of the last step of the equation it results in the distance formula.
Using the Pythagorean Theorem to derive the Distance Formula
Next we had to use this theorem to prove the distance formula by using A squared + B squared = C squared, and change our triangle with the coordinates a and b. Then you have the Formula as seen on the left.
Using the Distance Formula to derive the equation of a circle centered at the origin of a Cartesian coordinate plane and Defining the Unit Circle
As you can see once we figure one thing out we use it to find the equation or formula to find the next thing. Now that we have the Distance Formula we can use it to find the equation of a circle centered at the Cartesian point also called a unit circle. The circle has a radius of one so it makes it a easy reference point. All points of any circle, including the unit circle are equidistant from a given point. In mathematics, a unit circle is a circle with a radius of one. The unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. We also proved that all circles are similar because of this and can be dilated to prove its similarity. The equation for this is x squared + Y squared =1.
Finding points on the unit circle (at 30 degrees, 45 degrees and 60 degrees)
Now that we know this to find the points on the unit circle at 30, 45, and 60 degrees we just follow the same steps and use the formula X squared + Y squared =1. When solving for 45 degrees, we can see the x and y are equal and keep 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 because it's being reflected. This gives us enough information to finish solving alike to the problem previously. 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.
Using the symmetry of a circle to find the remaining points on the unit circle and defining the tangent function
Now we get to finding the remaining points of a circle. Sounds hard but luckily we know the unit circle symmetry. With that we can calculate the location of the unknown points by reflecting the known points to their position. Also with using the unit circle we can make proof of the tangent function by reflecting it upon the x-axis and then it shall overlap perfectly, therefore proving it is 90 degrees
Using the unit circle to define sine and cosine (of the angle theta), Defining the tangent function and Using similarity, proportions to derive the general trigonometric functions (sine, cosine and tangent) andDefining the tangent function
Now back to triangles we learned about some important trigonometric formulas for using a shape as simple as a right triangle. First labeling the triangle in relation to where the delta angle is located helps to show which lengths you will working with, the sides are labeled: opposite, adjacent, and hypotenuse(I label them O,A,H). We learned about sine by understanding 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. Next is tangent, written as T=O/A. These key trigonometric terms were learned by dissecting the unit circle into triangles that allowed me to look deeper into the intercepts and angles to solve for a specified point.
Using similarity and proportions to derive the general trigonometric functions (sine, cosine and tangent), Using the unit circle to define the arcsine, arccosine and arctangent functions, Using the Mount Everest problem to discover the Law of Sines ("taking apart"), and Deriving the Law of Sines Deriving the Law of Cosines
To finish it off I will be talking about a next set of terms we covered: Arc sine, Arc cosine, Arc tangent. These can each be derived by finding the reverse or opposite of either sine, cosine, or tangent. Knowing that the opposite of a positive will always be its negative: angle theta=cos^-1 or sine and tangent. We learned about the Law of Sine by working with triangles that were not right triangles, meaning that the Pythagorean theorem could not be applied. One problem we solved using this method was the Mount Everest problem, in which we split into to right triangles by dropping a perpendicular and then solving for the missing length. We were able to use this method because we knew two of the angles and one of the side lengths given. Written out the law of sine is sin B/b = sin C/c = sin A/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.