Discoveries in astronomy made during the Renaissance Era that involved trigonometry were made by Copernicus. Copernicus dedicated his work to creating a new model of the solar system to replace Ptolemy’s, except with the sun at the center and not the Earth. He achieved this with the use of trigonometric functions to calculate the distance between planets and the sun using an astronomical unit that he did not have a numerical value for. At the time, it was accepted that a planet’s orbit was in the shape of a perfect circle. The concept of the unit circle was not fully developed until the Cartesian coordinate system was invented by Rene Descartes, but applications of angles and arcs were already developed by the Islamic empire. Copernicus used a heliocentric solar system model and then the known locations of stars to calculate the positions of every known planet’s orbit. By applying the relationships between angles and arcs, Copernicus could determine the distances between each planet’s orbit relative to his astronomical unit, even though it didn’t have a numerical value. Copernicus also made use of spherical trigonometry, which has slightly different identities than plane trigonometry. For example, in spherical trigonometry, the cosine identity is cosa = cosbcosc + sinbsinccosA. In spherical trigonometry, a,b, and c are the side lengths of a spherical triangle, but these side lengths are equal to the angles created by those side lengths from the center of the sphere, and A is the angle of the spherical triangle opposite of side a. Almost every identity in spherical trigonometry is derived from the cosine identity, which Copernicus wrote about in his 1542 publication On the Sides and Angles of Triangles.
Many aspects of Copernicus’s model hold true, except for the fact that planetary orbits are not perfect circles. That discovery, and the trigonometry involved with it, was made by Johannes Kepler. He determined that a planetary orbit is actually in the shape of an ellipse with the sun at its focus, not its center. Finding the foci, area, and circumference of an elliptical orbit involves a lot of trigonometry. An ellipse is best described as a conic section. While a circle in a cone would cut perpendicular to the height of the cone, an ellipse does not. In order to find the area and the distances of the foci of an ellipse, the trigonometric functions sine and cosine are used. As seen in the diagram above, an ellipse’s foci create a triangle with a base along the major axis. For an orbit, the two points on either side of the major axis are the aphelion and the perihelion. The aphelion is the point farthest from the sun and the perihelion is the point closest to the sun. The lengths of the two sides of the triangle stretching from the planet to the foci are typically denoted as r and r’, where r is the shorter side. The minor axis is the height of the ellipse. To calculate r and r’, the relationship r + r’ = 2a is used, where a is half of the major axis. To find r, the equations r=a(1-e2)1+ecos or r’2=r2+4ae(ae+rcos) are implemented, where 𝜃 is the angle created by r and the major axis opposite the arc and e is the eccentricity of the ellipse, or how elongated it is and is equal to 1 – (perihelion)/a. The area of the ellipse can be calculated by using trigonometry with the equation Area=2ab0/2(1+cos2u)du=2abu-sin2u20/2=ab where a and b are the variables from the Cartesian equation for an ellipse x2a2+y2b2=1. Kepler’s other great contribution to planetary orbits is Kepler’s Equation, which tells the relationship between the mean anomaly M and the eccentric anomaly E of an elliptical orbit. The mean anomaly is the angle of the orbit from the center of the sphere and is equal to M(0)+360°(t/T), where t is the time that has passed and T is the period of the orbit. When drawing the orbit of a satellite, a circle can be drawn perpendicular to the ellipse. Given the position of the satellite on the ellipse, a corresponding point can be drawn on the circle in line with the satellite. In the diagram to the left, ellipse A is the orbit with point C at its center, circle B is the perpendicular circle, point P is the satellite, point Q is the corresponding point on the circle, and line D represents how Q and P are lined up, having the same domain on a cartesian plane. Kepler’s equation states that M=E-(180°/)e sinE.
Another great discovery regarding astronomy made with trigonometry before the modern era is James Bradley’s discovery of Earth’s nutation and stellar aberration. Up until Bradley’s discovery, astronomers had simply accepted Copernicus’s and Kepler’s notions that the Earth moved in an orbit around the sun, but none had attempted to prove it. Stellar aberration is the apparent movement of stars due to the movement of Earth and nutation in astronomy is a concept in which the speed of a planet in orbit is not constant. Bradley was able to prove both by comparing the angles made by Earth’s position and the observed position of a specific star called Eltanin. In order to notice the difference between the expected angle and the observed angle, Bradley based his expectations on what was already known about stellar parallax. Stellar parallax is a concept that states, since a star has a fixed position, the observed position of a star relative to Earth should be opposite of the Earth’s position in its orbit. What Bradley found, though, is that the star’s observed position was not offset by Earth’s position but rather by Earth’s velocity, proof that Earth was moving in a circular orbit. Bradley stated the relationship between these angles of displacement as sin(1)=sin(1)vJanuaryc using the law of sines, with the same relationship true for 2, 2, and vJuly. He was also able to calculate the speed of light within 0.5% of Einstein’s modern-day accepted calculation. Bradley attributed the observed stellar aberration of stars due to the behavior of light particles. The reason Bradley had studied Eltanin was because it passed directly over London at a certain time during the year, so by starting on that date and then finding the angle he had to point his telescope at for certain periods of time afterwards gave him the angle of displacement. Because of how light behaves, Bradley determined that the reason for that angle was because of Earth’s motion. Light will bend in the direction of an object’s motion much like how rain that is falling directly downwards will appear to be falling at an angle when you are on a moving train. Bradley calculated that the angle between the vector of the actual direction of a star and the apparent direction of a star is =v/c in radians, v being the velocity of the Earth and c being the speed of light. Using this Bradley calculated the speed of light to be 301,000 km/s. The actual value is 299,792 km/s.
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