As my science fair topic, I chose to test the accuracy of using parallax to measure distance. I chose this topic because it relates to two of my favorite topics: mathematics and astronomy. Parallax uses a mathematical formula and is most commonly used to measure the distance between celestial bodies. From my research on parallax, I found how to measure it, and how to use the parallax formula to measure distances.
Parallax is defined as “any alteration in the relative apparent positions of objects produced by a shift in the position of the observer” (Columbia Electronic Encyclopedia 1). Parallax is commonly used to measure distances between celestial bodies, such as planets and stars. Parallax is measured using angles that are much smaller than a degree. Arcminutes are one sixtieth of a degree and arcseconds are one sixtieth of an arminute. One example of the infinitesimal size of an arcsecond could be the width of a dime from a point of view two kilometers away (“Cool Cosmos”). These units of measurement are used in the parallax formula, or the formula used to calculate distance when given an object’s parallax measurement. The distance given from the parallax formula is in parsecs, which are 3.26 light years or 3.18x10^13 kilometers (“PARALLAX”). The parallax formula can be written as “distance = 1/parallax” (“PARALLAX”).
The parallax formula is derived using trigonometric functions in relation to right triangles and parallax angles. “The Six Trigonometric Functions and Reciprocals” says the six basic trigonometric functions are sine, cosecant, cosine, secant, tangent, and cotangent. In a right triangle, the sine of an angle is the opposite side from the angle divided by the hypotenuse of the triangle. Cosecant is its reciprocal. The cosine of an angle is the side adjacent to the angle divided by the hypotenuse. Secant is its reciprocal. The tangent of an angle is the side opposite of the angle divided by the side adjacent to the angle. Cotangent is its reciprocal (“The Six Trigonometric Functions and Reciprocals”).
“Deriving the Parallax Formula” shows that one way of deriving the parallax formula is to set up a right triangle consisting of Earth, the Sun, and one other star as vertices. The side going from Earth to the Sun can be labeled as “a” and the side from the Sun to the other star can be labeled as “d.” The angle between the other star and Earth can be labeled as “p.
Trigonometric ratios are something you would hope to never use, but this term I was forced to find the height of a given eucalyptus tree. No instructions were given for this task other than the fact that I was not allowed physically measure the tree itself. The tools I was given were a large protractor and a measuring tape. I immediately sat down and thought why we were given the specific tools. I soon came up with a theory knowing that If I can find an angle that lines up with the top of the tree (θ) and the adjacent side (length
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