Overview and Basic Definitions
Trigonometry is arguably one of the most essential and applied branches of mathematics. From generating graphic representations of sound waves through sine and cosine functions to measuring the height of a mountain using the Law of Sines, trigonometry is an integral part of the world we live in today. It was a concept first brought about by the ancient Greeks (as well as Babylonians, but mostly Greeks) , specifically three main scholars named Hipparchus, Menelaus, and Ptolemy. Hipparchus of Nicaea was the first scholar known to have used a trigonometric table, which he used for predicting the eccentricities (the measures of how much celestial bodies deviate from a circular path) of the orbits of the sun and moon and is also credited with tabulating values for the chord function. Menelaus of Alexandria is most known for his work called Sphaerica, which deals with the geometry of the sphere, particularly the concept of the spherical triangle (figures formed of three circle arcs) also known as a trilateral. Ptolemy's most famous work on trigonometry is known as the Almagest, which focuses on plane and spherical trigonometry. Of course, the sine, cosine, and tangent functions we learn in school today are more recent concepts, but the general idea that it is possible to measure the world around us using trigonometry came about thousands of years ago.
Some of the most basic concepts of trigonometry that we covered in class are the sine, cosine, tangent, ArcSine, ArcCosine, and ArcTangent functions, the Law of Sines, and the Law of Cosines. The sine is a function that represents the ratio of the side opposite to a given angle of a right triangle to the hypotenuse, while the cosine function is one that represents the ratio of a side adjacent to a given angle in a right triangle to the hypotenuse, and the tangent of an angle is the length of the opposite side of a right triangle divided by its adjacent side. The Law of Sines is a trigonometric law which 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, and the Law of Cosines is another trigonometric law used primarily to find the remaining parts of an oblique triangle when either two side lengths and an angle measure are given or if all three side lengths are known (SAS or SSS).
Some of the most basic concepts of trigonometry that we covered in class are the sine, cosine, tangent, ArcSine, ArcCosine, and ArcTangent functions, the Law of Sines, and the Law of Cosines. The sine is a function that represents the ratio of the side opposite to a given angle of a right triangle to the hypotenuse, while the cosine function is one that represents the ratio of a side adjacent to a given angle in a right triangle to the hypotenuse, and the tangent of an angle is the length of the opposite side of a right triangle divided by its adjacent side. The Law of Sines is a trigonometric law which 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, and the Law of Cosines is another trigonometric law used primarily to find the remaining parts of an oblique triangle when either two side lengths and an angle measure are given or if all three side lengths are known (SAS or SSS).
Proving the Pythagorean Theorem
In class, we established the fact that in order to find the length of the hypotenuse of a right triangle, one must use the Pythagorean Theorem, or the equation known as a squared + b squared = c squared. In order to verify that this theorem is the way to find the hypotenuse of a triangle, we had to first solve a proof. In mathematics, a proof is an argument over a previously established mathematical statement, such as a theorem. We solved the above dissection of the Pythagorean theorem proof when we did the worksheet "Proof By Rugs" as shown in the images below. The area of the first square can be represented as the equation 4(1/2ab)+a squared +b squared, and the area of the second square can be represented by another equation; 4(1/2ab)+c squared. Then, we set both of these equations equal to each other since they have the same areas. The equations would then be shown as 4(1/2ab) + a squared + b squared = 4(1/2ab) + c squared. If we subtract the equals from both sides of this equation, we will come up with a squared + b squared = c squared. Thus, we proved that the area of a square on the hypotenuse of a right triangle is the sum of the areas of the squares on its legs, or in other words, that a squared + b squared = c squared.
Deriving the Distance Formula with the Pythagorean Theorem
In class, we learned that the distance formula is the square root of (x1-x2) squared + (y1-y2) squared as well as how to derive it from the Pythagorean Theorem, which we briefly revisited. In order to derive it, we drew a right triangle on a coordinate plane and labeled two of the points as (x1, y1) and (x2, y2) and the hypotenuse as d for distance. We now know that the legs of the triangle are (x1-x2) and (y1-y2). To find d we must use the Pythagorean Theorem, so the values plugged into the equation equal d squared = (x1-x2) squared + (y1-y2) squared. Then, to find the answer for d, we drew a radical sign over the original equation, thus finding the distance formula known as the square root of (x1-x2) squared + (y1-y2) squared.
Deriving the Equation of a Circle on a Cartesian Coordinate Plane Using the Distance Formula
As a polygon increases its number of sides, it becomes more and more similar to a circle. A circle, as we learned in class, is the locus of all points that are equidistant from a single central point. The premise of this lesson was to derive the equation of a circle (x squared + y squared = 1) using the distance formula which we had derived earlier. In order to accomplish this, we had to graph a circle on a Cartesian coordinate plane which was centered on the origin (0,0). We can assume that the radius of this circle is 1 and that (x,y) on the coordinate plane above is the ordered pair (3,2). Knowing the distance formula, we can calculate the side lengths adjacent and opposite to the hypotenuse.
Defining the Unit Circle
The unit circle, especially in trigonometry, is a circle with a radius of one unit that is centered at the origin on a Cartesian coordinate plane, such as the image depicted above.
Finding Points on a Unit Circle (at 30 degrees, 45 degrees, and 60 degrees)
Since we learned about the unit circle in class, our next task was to find the points on it, specifically at the 30, 45, and 60 degree points. For the task of finding a point at the 30 degree area, we extended the radius of the circle from the origin at a 30 degree angle. As we know, this radius is one unit. Now, we drop a perpendicular line from the point of the radius that touches the circumference of the circle, which makes a right triangle with a hypotenuse of one unit. We labeled the legs of the triangle x and y, with y being the shortest leg of the triangle. In order to find the other side lengths of the right triangle, we reflected it across the x-axis which made an equilateral triangle with all three sides having a side length of one unit. This makes the side length y equal to 1/2, which then establishes 1/2 as the y-coordinate in the ordered pair we were seeking to find. Now that we know the hypotenuse and one of the legs of the triangle, we simply use the Pythagorean Theorem to find the other side length. As shown in the image to the right, we came up with the answer known as the square root of 3/4, which simplifies to the square root of 3/2, which becomes the x-coordinate of the ordered pair.
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To find the point at the 60 degree area, we extended the radius (which is still one unit) of the circle from the origin at a 60 degree angle. When we drop a perpendicular line from the point that touches the circumference, a right triangle is formed, specifically a 30-60-90 triangle. When we reflect it across the x-axis, an isosceles triangle forms. The two equal sides of this triangle are, as we know, one unit each, and the height of this triangle is 1/2 because we found out from the last task that the length of the shortest leg of the triangle is 1/2. This is now the x-coordinate of the ordered pair. In order to find the remaining side length, one must simply calculate the cosine of 30 degrees or cos (30). The answer to cos (30) is the square root of 3 over 2, which becomes the y-coordinate in the ordered pair.
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To find the ordered pair for a point at the 45 degree area of the unit circle, we used the cosine function to find the x-coordinate on the ordered pair and the sine function to find the y-coordinate on the ordered pair. Cos (45) equals the square root of 2/2 for the x-coordinate and sin(45) equals the square root of 2/2 for the y-coordinate.
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Using Symmetry to Find the Remaining Points on the Unit Circle
Once we have found the points of the unit circle in the first quadrant of the coordinate plane, we can use symmetry to find the rest of the points on the circle. For example, If we were to take the point (square root of 2/2, square root of 2/2) and reflect it across the y-axis into Quadrant II, the new point would be called (- square root of 2/2, square root of 2/2) because it is the product of reflectional symmetry. Likewise, If we took the original point and instead reflected it across the x-axis into Quadrant IV, the new point would be known as (square root of 2/2, -square root of 2/2). In this way, we can find all the points in a unit circle with the help of reflectional symmetry.
Using the Unit Circle to Define Sine and Cosine (of Theta)
In terms of the unit circle, the cosine of theta can be defined as the x-coordinate of the intersection point of the unit circle and a radial line that is at an angle of theta to the positive x-axis, and the sine of theta can be defined as the y-coordinate of the intersection point of the unit circle and a radial line that is at an angle of theta to the positive y-axis.
Defining the Tangent Function
The trigonometric definition of the tangent of an angle is the length of the opposite side divided by the length of the adjacent side abbreviated as tan.
Using Similarity and Proportions to Derive General Trigonometric Functions
The sine, cosine, and tangent functions can all be derived using proportions in terms of right triangles. The cosine function equals the ratio of the side adjacent to the hypotenuse to the hypotenuse itself. It is represented as cos (theta) = A/H (CAH). The sine function equals the ratio of the side opposite the hypotenuse to the hypotenuse itself. It is represented as sin (theta) = O/H (SOH). Lastly, the tangent function equals the ratio of the side opposite the hypotenuse to the side adjacent to the hypotenuse. This is represented as tan (theta) = O/A. These three basic functions together form the mnemonic SOHCAHTOA, which simply describes all the functions.
Using the Unit Circle to Define the arcSine, arcCosine, and arcTangent Functions
The arcSine, arcCosine, and arcTangent functions are essentially the inverse of the sine, cosine, and tangent functions. They are used to get an angle from any of the angle's trigonometric ratios. The arcSine function is represented as theta = arc sin (o/H), the arcCosine function as theta = arc cos (A/H), and arcTangent as theta = arc tan (O/A).
Using the Mount Everest Problem to Discover the Law of Sines
In the Mount Everest problem, we learned about how the British conducted the Great Trigonometrical Survey in order to calculate the height of Mount Everest. They triangulated the location of the peak using massive instruments called theodolites. Using only three pieces of information, the British were able to triangulate the peak's exact location without crossing the borders of any countries. The three pieces of information they had to find the distance were two angles of the triangle and one side length. Our task was to find the distances of the other two side lengths of the oblique triangle.
Deriving the Law of Sines
ABC is the triangle and h in this case is known as the altitude, or height, and let ADC represent the new right triangle formed by drawing the altitude. In triangle ADC, the sine is represented as sin A = h/b, so this means that h = b sinA. In triangle ABD, sin B = h/a, so h = a sinB. Now we know that b sinA = a sinB, so when we divide both sides by sinA and sinB, we get b/sinB = a/sinA, so it only remains to be said that a/sinA = b/sinB = c/sinC.
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