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Trigonometry is the discipline of mathematics that studies the relationship between the sides and angles of a right-angled triangle. The listed trigonometric ratios, which are mentioned such as sine, cosine, tangent, cotangent, secant, and cosecant, are employed to examine this relationship. The concept of trigonometry was established by the Greek mathematician Hipparchus, while the word trigonometry is a 16th century Latin derivative.
We will learn about the fundamentals of trigonometry, the many identities-formulas of trigonometry, and real-life instances or applications of trigonometry in the sections below.
Trigonometry Basics:-
The measurement of angles and issues involving angles are covered in trigonometry basics. Trigonometry has 3 basic functions: sine, cosine, and tangent. Other essential functions can be derived using these three basic ratios or functions: cotangent, secant, and cosecant. These functions form the foundation for all of the fundamental trigonometry topics. As a result, the order to comprehend trigonometry, we must first study these functions and their formulas.
The three sides of h right-angled triangle are as follows.
The side opposite the angle is called perpendicular.
The angle’s neighbouring side is called the base.
The hypotenuse is the side of the right angle that is opposite the right angle.
Ratios Trigonometric
In trigonometry, there are six fundamental ratios.
- sinA=perpendicular/hypotenuse
- cos A= hypotenuse/base
- tanA = base/perpendicular
The values of the other three functions, cot, sec, and cosec, are determined by the values of tan, cos, and sin, as shown below.
- cotA = 1/tan A= base/perpendicular
- sec A= 1/cos A= hypotenuse/base
- cosec A = 1/sin A= hypotenuse/perpendicular cosec = 1/sinA
Trigonometric angles are the angles in a right-angled triangle that can be used to illustrate various trigonometric functions. In trigonometry, standard angles include 0° ,30°, 45°, 60°, and 90°. The given angles’ of trigonometric values can be found directly in a trigonometric table. 180°, 270°, and 360° are some more key angles in trigonometry. The angle of trigonometry can be stated using trigonometric ratios as follows:
sin-1A= Hypotenuse/Perpendicular
cos-1A=Base/Hypotenuse
tan-1 A=Base/Perpendicular
Solved trigonometry examples:-
Question 1. The elevation angle of the top of the building at a distance of 50 m from its foot on a horizontal plane is found to be 60°. Find the height of the building.
Solution:-
The letters AB and BC reflect the building’s height and distance from the point of observation, respectively.
The opposite side (AB) of the right triangle ABC is known as the opposite side (AB), the hypotenuse of given figure is the side (AC) is known as the hypotenuse side (AC), and the remaining side is known as the adjacent side (AC) (BC).
Now we must determine the length of side AB.
tan = Adjacent side/opposite side
AB/BC = tan60°
AB/50 = √3
AB = √3 x 50
3 has an approximate value of 1.732.
50 = AB (1.732)
AB = 86.6 metres
As a result, the building’s height is 86.6 feet.
Q2. A guy observes the base of a tree at a 30° angle of depression from the top of the 30 m tall tower. Determine how far the tree is from the tower.
Solution:-
The height of the tower is represented by AB, while the distance between the tower’s foot and the tree’s foot is represented by BC.
Now we must determine the distance between the tower’s foot and the tree’s foot (BC).
opposite side/adjacent side = tanA
AB/BC = tan30°
30/BC = 1/√3
30×√3 =BC
3 has an approximate value of 1.732.
51.96 metres
As a result, the tree and the tower are 51.96 metres apart.
Q3. A ladder leaning against a wall that reaches the top of a 6-meter-high wall and is slanted at a 60-degree angle. Determine how far the ladder is from the wall’s foot.
Solution;
The letters AB stand for the wall’s height, BC for the distance between the wall and the ladder’s foot, and AC for the ladder’s length.
The opposite side of the right triangle ABC is known as the opposite side (AB), the hypotenuse side is known as the hypotenuse side (AC), and the remaining side is known as the neighbouring side (BC).
Now we must determine the distance between the ladder’s foot and the wall. That is, we must determine BC’s length.
opposite side/adjacent side = tanA
AB/BC = tan60°
3 Equals 6 (BC)
6/√3 = BC
(6/√3) x (√3/√3) = BC
(6×3)/√3 = BC
√3 has an approximate value of 1.732.
BC = 2×(1.732)
3.464 metres BC
What is trigonometry:
Trigonometry is a discipline of mathematics dealing with the application of certain angles functions to calculations. In trigonometry, there are six functions of an angle that are often utilised. Sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant are their names and acronyms (csc). The image depicts these six trigonometric functions with respect to a right triangle. The ratio of the side opposite A to the side opposite the right angle (the hypotenuse), for example, is called the sine of A, or sin A; the other trigonometric functions are defined similarly. These functions are angle A attributes that are independent of triangle size, and determining values. The word trigonometry is derived from the Greek words for the triangle (trignon) and measure (metry) (metron). Though the field began in Greece in the third century B.C., it was India in the fifth century A.D. that made some of the most significant contributions (such as the sine function). Due to the loss of early Greek trigonometric writings, it is unknown whether Indian scholars created trigonometry independently or as a result of Greek influence. Trigonometry arose primarily from the needs of Greek and Indian astronomers, according to Victor Katz in “A History of Mathematics (3rd Edition)” (Pearson, 2008). listed for numerous angles.
Conclusion :
In this article, we have dealt with various trigonometric functions which will help you to clear the concept of trigonometry. In this article we come across very basic trigonometry then we have seen variously trigonometry examples and then the most important the meaning of trigonometry.
