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Trigonometry expands on the idea of a right triangle to teach pupils the fundamentals of trigonometry. The name ‘trigonometry’ comes from the Greek words ‘trigonon’ which means the triangle and the word ’metron’ which means measurement or meaning measure. The study of relationships between the sides and angles of a triangle is known as trigonometry. Trigonometric ratios and trigonometric identities are the two key ideas that students will master in this chapter. Trigonometric concepts are used in the majority of technologically advanced procedures used in Engineering and Physical Sciences, making it a highly significant topic.
Meaning of Trigonometry
The measurement of the angles as well as problems involving angles are covered by the principles of trigonometry. The three fundamental functions of trigonometry are sine, cosine, and tangent and all other essential trigonometric functions can be built using these three basic ratios or functions: cotangent, secant, and cosecant. These functions serve as the foundation for all of trigonometry’s important issues. As a result, we must first study these functions and their formulations in order to understand trigonometry.
The following are the three sides of a right-angled triangle. Perpendicular refers to the side that is perpendicular to the angle’ A ‘. The base is the side of angle ‘A’ that is adjacent to it.
The hypotenuse is the side of the right angle which is opposite to the right angle.
Students must review all of the ideas from the previous three exercises in order to solve NCERT solutions class 10 mathematics chapter 8 exercise 8.4. It’s worth noting that we can express each trigonometric ratio in terms of other trigonometric ratios using these identities, i.e., if the values of any one of the ratios are known, we may also derive the values of other trigonometric ratios.
The six most important trigonometric functions which are also known as trigonometric ratios are derived using the formulas below . Because they define a set of important trigonometric functions, the sides of the right triangle are crucial to comprehend.
SinA = P / H
CosA = B / H
TanA = P / B
CosecA = H / P
SecA = H / B
CotA = B / P
Where P refers to the perpendicular , B refers to the base and H refers to the height of the right-angled triangle.
Trigonometric expressions
Many formulas in trigonometry show the links between trigonometric ratios and angles for different quadrants. A list of basic trigonometric formulas is given below:
1. Trigonometric ratios
SinA = P / H CosA = B / H
TanA = P / B CosecA = H / P
SecA = H / B CotA = B / P
Where P refers to the perpendicular , B refers to the base and H refers to the height of the right-angled triangle.
2. Trigonometric formulas using Pythagoras’s identity
Sin²A + cos²A = 1
1 + tan²A = sec²A or sec²A – tan²A = 1
1 + cot²A = cosec²A or cosec²A – cot²A = 1
3. Trigonometry’s Sine and Cosine Law
a/sinB = b/sinB = c/sinC
2ab cos C = a² + b² – c²
2bc cos A = b² + c² – a²
2ac cos B = a² + c² – b²
The lengths of the triangle’s sides are a, b, and c, and the triangle’s angle is A, B, and C which is opposite to the sides a , b , c .
Example 1 . Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.
Solution :- As we know that cosec²A – cot²A = 1 then
cosec²A = 1 + cot²A and 1 / sinA = cosecA
so 1/sin²A = 1 + cot²A => sin²A = 1/ 1 + cot²A
And sinA = √ 1 / 1 + cot²A
We know that CosA / SinA = cotA so CosA = SinA . CotA
- CosA = (√ 1 / 1 + cot²A) . CotA
- SecA = CotA . 1/(√ 1 / 1 + cot²A)
And tanA = 1 / cotA
Conclusion :-
Trigonometry is a very important subject of mathematics with a wide range of applications. The relationship between the sides and angles of a right-angle triangle is studied in the subject of mathematics known as “trigonometry.” As a result, missing or unknown angles or sides of a right triangle can be found using trigonometric formulas, functions, or identities. In trigonometry, angles are expressed in degrees or radians.
Trigonometric ratios are the ratios between the edges of a right triangle. These ratios are obtained by applying the following trigonometric functions to the specified angle A, where a, b, and c correspond to the lengths of the sides. 0⁰, 30⁰, 45⁰, 60⁰, and 90⁰ are some of the often utilized angles.
