Mathematics has the use of trigonometric functions. The trigonometric functions are also referred as the goniometric functions, angle functions or circular functions. The primary goal of these functions are to relate two-side lengths ratio of a right-angled triangle to an angle. All the field of science that are related to the domain geometry has the use of these functions. Such examples of fields are celestial mechanics, solid mechanics, geodesy, navigation and others. The functions used are in simple form and sometimes used to study the periodic phenomena with the help of Fourier analysis. The most important terms and functions of the trigonometric functions are known as the sine function, cosine function and the tangent function. The reciprocal or the multiplicative inverse of these functions are the cosecant function, secant function and the cotangent function. These function are used in the field of the mathematics. The oldest trigonometric definitions are relating to the acute angles of the right-angle triangle.
There are a total of six trigonometric functions available and the name of those functions are - Sine (sin), Cosine (cos), Tangent (tan), Secant (sec), Cosecant (csc) and the last one is Cotangent (cot). In a right-angled triangle, there is one particular angle with the value of 90o and there are two other angles with the value less than 90o each. These angles are known as the acute angles. Every right-angled triangle has a hypotenuse and to determine which side is the hypotenuse we have to see that which side is connecting the two angles, which are acute. In simple words, the hypotenuse is the side which is exactly opposite of the angle which has the value of 90o. The side of the triangle, which parallel to the horizon is known as the adjacent. The base is usually the side, which is at the bottom. The side, which is the third side, is known as the opposite. If the value of an acute angle is known then all the sides of the right-angled triangle are defined. Up to a factor of scaling these sides are well defined. An acute angle has six functions depending on the ratios of the sides. An acute angle determines all the ratios of any two sides. These are the functions of the trigonometry. The core functions of the trigonometry are the Sine, Cosine and Tangent. Before proceeding further in to the trigonometry one need to know about these functions.
Sine (sin): Sine theta is the ratio of opposite / hypotenuse.
Cosine (cos): Cosine theta is the ratio of adjacent / hypotenuse.
Tangent (tan): Tangent theta is the ratio of opposite / adjacent.
There are other functions, which are not commonly used. These functions can be derived with the help of the above-mentioned primary functions. As these functions can be derived easily with the help of primary functions, there is no mention of these functions when people use any type of spreadsheets or calculator.
Secant (sec): Secant theta is the ratio of hypotenuse / adjacent.
Cosecant (csc): Cosecant theta is the ratio of hypotenuse / opposite.
Cotangent (cot): Cotangent theta is the ratio of adjacent / opposite.
The Cosecant theta can derived from the primary function Sine theta. Dividing 1 with the value of sine theta will give the result as Cosecant theta. The Secant theta can be derived from the primary function Cosine theta. Dividing 1 with the value of Cosine theta will provide the result as the Cosecant theta. Cotangent theta can be derived from the primary function Tangent theta and to have the Cotangent we have to divide 1 with the value of the Tangent theta.
There are few steps, which is to be known while solving the problems in trigonometry. Step 1: We have to know the formulas of the all trigonometry functions, which includes the formulas of main trigonometry functions. We have to be familiarized with all the terms that we use in trigonometry. The definitions of the Hypotenuse, Opposite, Right Triangle and Adjacent should be known.
Step 2: Once we know all the terms and the definition. We have to find the value of sine, cos and tan of the triangle consisting the specific angles. The values of the sides need to be known.
Step 3: All the tan, cot, sec, csc, need to be converted to sin and cos. All the angles need to be checked for differences and sums. Those need to be removed with the help of proper identities.
Step 4: Angle multiple need to be checked and removed with the help of formulas.
Step 5: All the equations need to be expanded and simplified.
Step 6: If any cos function, powers more than 2 then that need to be replaced with sine powers with the help of Pythagorean identities.
Step 7: Denominators and numerators need to be factored and common factors need to be canceled.
Step 8: At this point, the two sides should be equal and the identity is proven.
There are many real-life examples of trigonometry such as –
The functions of trigonometry can be used to measure or calculate the height a mountain, building or pole. Assume the distance of a pole from a certain viewpoint is 90ft. and the angle of the elevation to the pole top is 35o. Now to find the pole height we have to solve the problem with the help of trigonometry. In this problem, the values of the adjacent side and the acute angle is given. We have to find the value of the opposite site. Hence, we have to use the function of Tan.
Tan 35o=opposite side/adjacent side.
Tan 35o = H / 90.
H = 90 × tan35o
H = 90 × 0.4738
H = 42.64 feet
From the above calculation, the height of the pole is determined.
Trigonometry has the use in video games as functions of trigonometry have an important role. Construction of buildings have also the use of trigonometry as the height, width, structural roof slopes and many other factors requires the application of trigonometry. Trigonometry has also use in the field of flight engineering, physics, marine biology, marine engineering and navigation.
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