All About Trigonometry.
~What is Trigonometry ~
You may have learned about trigonometric functions such as sine and cosine as being defined by the ratios of sides of a triangle, or in terms of points and lines related to the unit circle.
For example, you can think about the sine function as measuring the distance from the x-axis of a point on the unit circle at a particular angle. The sign (+/-) of that value indicates if the point lies above or below the axis. Similarly, the cosine can be thought of as measuring the distance from the y-axis of that same point.
It is useful to note that the cosine of an angle is the same as the sine of the complement of the angle. In other words, it is the same operation as sine, just with respect to the y-axis instead of the x-axis.
The word sine originally came from the Latin sinus, meaning “bay” or “inlet”. However, it had a long path to get there. The earliest known reference to the sine function is from Aryabhata the Elder, who used both Ardha-jya (half-chord) and jya (chord) to mean sine in Aryabhatiya, a Sanskrit text finished in 499 CE.
Jya, meaning chord, became jiba in Arabic and was abbreviated as just jb. When the term was translated to Latin in the twelfth century, jb was incorrectly read as jaib (meaning “bay” or “inlet”), and thus translated as sinus.
Tangent comes from the Latin tangere, the verb meaning “to touch”. A line tangent to a circle intersects it at exactly one point. From this, geometric construction of the tangent function makes a lot of sense: take the line tangent from a point on the unit circle and calculate the distance along that line from the point of intersection with the circle, to the point of intersection with the x-axis. Similarly, the distance from that same point on the unit circle to the y-axis is the value of the cotangent function.
The origin of secant can be traced to Latin as well. It comes from the Latin word secure, meaning “to cut”. By definition a secant line on a circle is any line that intersects it in two places, you can think of this line as cutting the circle in two pieces. The secant function is the distance from the origin to the point where the tangent line intercepts the x-axis. Note that if this secant line is extended, it cuts the unit circle neatly in half. Again, the cosecant can be thought of as being the same as the same function with respect to the y-axis.
More Info About Trigonometry
~History Of Trigonometry ~
Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics (Rhind Mathematical Papyrus) and Babylonian mathematics. Trigonometry was also prevalent in Kushite mathematics.[1] Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic astronomy.[2] In Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata (sixth century CE), who discovered the sine function. During the Middle Ages, the study of trigonometry continued in Islamic mathematics, by mathematicians such as Al-Khwarizmi and Abu al-Wafa. It became an independent discipline in the Islamic world, where all six trigonometric functions were known. Translations of Arabic and Greek texts led to trigonometry being adopted as a subject in the Latin West beginning in the Renaissance with Regiomontanus. The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 17th-century mathematics (Isaac Newton and James Stirling) and reaching its modern form with Leonhard Euler (1748).
~Trigonometry Applications in Real Life~
It may not have direct applications in solving practical issues but is used in various fields. For example, trigonometry is used in developing computer music: as you are familiar that sound travels in the form of waves and this wave pattern, through a sine or cosine function for developing computer music. Here are a few applications where trigonometry and its functions are applicable.
Trigonometry to Measure Height of a Building or a Mountain
Trigonometry is used in measuring the height of a building or a mountain. The distance of a building from the viewpoint and the elevation angle can easily determine the height of a building using the trigonometric functions.
Trigonometry in Aviation
Aviation technology has evolved with many upgrades in the last few years. It has taken into account the speed, direction, and distance as well as the speed and direction of the wind. The wind plays a vital role in when and how a flight will travel. This equation can be solved by using trigonometry.
For example, if an airplane is traveling at 250 miles per hour, 55° of the north of east and the wind blowing due to south at 19 miles per hour. This calculation will be solved using trigonometry and find the third side of the triangle that will lead the aircraft in the right direction.
Trigonometry in Criminology
Trigonometry is even used in the investigation of a crime scene. The functions of trigonometry are helpful to calculate a trajectory of a projectile and to estimate the causes of a collision in a car accident. Further, it is used to identify how an object falls or at what angle the gun is shot.
Trigonometry in Marine Biology
Trigonometry is often used by marine biologists for measurements to figure out the depth of sunlight that affects algae to photosynthesis. Using the trigonometric function and mathematical models, marine biologists estimate the size of larger animals like whales and also understand their behaviors.
Trigonometry in Navigation
Trigonometry is used in navigating directions; it estimates in what direction to place the compass to get a straight direction. With the help of a compass and trigonometric functions in navigation, it will be easy to pinpoint a location and also to find distance as well to see the horizon.
Other Uses of Trigonometry
- The calculus is based on trigonometry and algebra
- The fundamental trigonometric functions like sine and cosine are used to describe the sound and light waves
- Trigonometry is used in oceanography to calculate heights of waves and tides in oceans
- It is used in the creation of maps
- It is used in satellite systems
More About Trigonometry
- Trigonometry is the mathematical study of the relations of the magnitudes of the sides and angles of right triangles (triangles with one 90° angle).
- Parts of triangles
- The location of the sides of any triangle is identified relative to an angle using the words opposite and adjacent.
- The sides of a right triangle adjacent to the right angle are called legs.
- The side of a right triangle opposite the right angle is called the hypotenuse.
- Standard position notation is used in this book
- Start with a diagram of a rectangular coordinate system.
- Label the “horizontal” axis pointing to the right of the diagram +x.
- Label the “vertical” axis pointing to the top of the diagram +y.
- Take any right triangle with one angle θ.
- Place the vertex of the angle θ on the origin.
- Align the leg adjacent to θ on the +x axis.
- Label the sides of the triangle.
- x is the leg adjacent to the angle (usually shortened to the adjacent).
- y is the leg opposite the angle (usually shortened to the opposite).
- R is the hypotenuse.
- The symbol R (or r) is used because the hypotenuse can be thought of…
- as the resultant formed by the addition x of and y.
- as the radius of a circle centered on the origin.
- Triangle inequality
- For any triangle, the sum of the lengths of any two sides is greater than or equal to the length of the remaining side.
- For a right triangle, the sum of the lengths of the legs is greater than or equal to the length of the hypotenuse.
- x + y ≥ R
- Pythagorean theorem
- For a right triangle, the square of the hypotenuse is equal to the the sum of the squares of the legs.
- R2 = x2 + y2
- The hypotenuse is always the longest side of a right triangle.
- Trigonometric functions…
- are ratios of the lengths of the sides of a right triangle
- as ratios, they are unitless (dimensionless)
- are defined relative to one of the angles that is not 90°
- often given the generic symbol θ (theta)
- come in six types, three of which are commonly used in physics…Commonly used trigonometric functions
