Trigonometry is a very important concept that has a wide application in the field of scientific and mathematical problem-solving. The term Trigonometry is obtained by adding two Greek words ‘Trigonon’ and ‘Metron ’ which means triangle and measure respectively. Trigonometry deals with the relationship between the different sides of a right-angle triangle. More specifically, it studies the ratios of the sides of a right-angle triangle and its angles other than the right angle. It is conveniently used to determine the unknown values of angles and sides of a right angle triangle using functions and formulas based on this relationship.
The ratios that are used to study the relationship in trigonometry are called trigonometric ratios or trigonometric functions. There are three basic functions in trigonometry. These are called sine, cosine, and tangent. These three basic functions or ratios are further used to derive another three trigonometric functions such as cotangent, secant, and cosecant. These ratios are usually denoted in short forms like sin, cost, tan, cot, sec, and cosec. All the important problem-solving techniques of trigonometry are being applied in mathematics through these six functions only.
These six basic functions play a very significant role in trigonometry to establish the relationship between the angle and the ratio of the sides of a right-angle triangle. If the angle formed by the hypotenuse and the base of a right angle triangle is x, then the trigonometric ratios can be denoted as follows:
sin x = Ratio of the opposite side of the angle to the hypotenuse = Altitude/Hypotenuse
cos x = Ratio of the adjacent side of the angle to the hypotenuse = Base/Hypotenuse
tan x = Ratio of the opposite side of the angle to the side adjacent to the angle = Altitude/Base
Now the other three functions are derived just by the inverse ratios of the above. These can be denoted as follows:
cot x = 1/tan x = Base/Altitude
sec x = 1/cos x = Hypotenuse/Base
cosec x = 1/sin x = Hypotenuse/Altitude
What is Unit Circle?
As per the definition, a circle is the locus of a point that is at a fixed distance from another point. The fixed point is called the center of the circle and the distance of the point from the center is called the radius of the circle. Any circle having a radius of one unit is termed the unit circle. So for the unit circle, any point on the circle’s edge is at a distance of one from the center. The center of the unit circle is usually considered as the point of intersection of the x-axis and y-axis in the Cartesian coordinate and is denoted by (0, 0).
The unit circle has all the properties of a circle and its equation can also be derived from the equation of a circle. The concept of the unit circle has some special and significant application in different areas in mathematics particularly in explaining the trigonometric functions. It helps to define the trigonometric functions like sine, cosine, and tangent.
Trigonometric Functions using Unit Circle
We can define the basic trigonometric functions with the help of a unit circle. A right-angle triangle is considered within the unit circle having the hypotenuse equal to the radius of the circle. So the hypotenuse has the value of 1 and the base and height of the right-angle triangle will be the x and y coordinates of the endpoint of the hypotenuse on the circle edge. The hypotenuse makes an angle θ with the positive direction of the x-axis. Then the trigonometric functions will be defined as follows:
Sin θ = Altitude/Hypotenuse = y/1
Cos θ = Base/Hypotenuse = x/1
Tan θ = Altitude/Base = y/x
These are the basic functions in trigonometry that express the relationship between the angle and the ratio of the height, base, and hypotenuse of a right-angle triangle. Cuemath is the best platform to learn more about the Unit circle.