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The study of trigonometry investigates how angles together with triangle side lengths relate within right triangles. Multiple practical applications rely heavily on trigonometry for their operation, including physics together with engineering and navigation. The chapter focuses on explaining trigonometric ratios of sine, cosine, tangent and their mathematical characteristics and demonstrates how these functions solve problems related to heights and distances.
Learning trigonometry leads to significant progress in advanced mathematics as well as to its useful applications throughout astronomy and architecture, and the technology field. Master Maths with a clear understanding of each chapter provided through the NCERT class 10th maths notes. And, expanding your comprehension through NCERT Notes, you should study related concepts from classes 9 to 12.
Trigonometry:
The word trigonometry is derived from the Greek words ‘tri’ (meaning three), ‘gon’ (meaning sides) and ‘metron’ (meaning measure). Trigonometry is the study of relationships between the
sides and angles of a triangle. Mainly, it deals with right-angled triangles, where one angle is always 90°.
In a right-angled triangle, there are three sides, which are known as follows:
These sides help define trigonometric ratios, which relate angles with sides.
Here, in the diagram below, ∠CAB (or ∠A) is an acute angle. Note the position of the side BC to ∠A. It faces ∠A. We call it the side opposite to angle A. Side AC is the hypotenuse of the right triangle shown below, and the side AB is a part of ∠A. So, call it the side adjacent to angle A.
The ratios of right-angled triangle sides relative to its acute angles form the basis of Trigonometric Ratios. The ratios serve to express how angle measurements relate to the lengths of triangle segments.
The six fundamental trigonometric ratios are:
Many practical problems in distances, heights and navigation, along with physics, require solutions by using ratios. The ratios serve as principles to understand both trigonometric identities and functions.
Many practical problems in distances, heights and navigation, along with physics, require solutions by using ratios. The ratios serve as principles to understand both trigonometric identities and functions.
In trigonometry, certain angles commonly appear in problems, such as 0°, 30°, 45°, 60°, and 90°. Trigonometric ratios of standard angles maintain constant values, which enable smooth solutions to mathematical and practical applications.
In triangle ABC, right-angled at B, if one angle is 45°, then the other angle is also 45°, i.e., ∠A = ∠C = 45°.
So, BC = AB (sides opposite to equal angles are equal)
Now, Suppose BC = AB =
Then by Pythagoras' Theorem, AC2 = AB2 + BC2
Hence, AC = √2
Using the definitions of the trigonometric ratios:
Sin 45° =
Cos 45° =
Tan 45° =
Cosec 45° =
Sec 45° =
Cot 45° =
Consider an equilateral triangle ABC. Since each angle in an equilateral triangle is 60°, ∠A = ∠B = ∠C = 60°.
Draw the perpendicular AD from A to the side BC.
From the Triangles Theorem, ABD ≌ ACD
And, BD = DC, ∠BAD = ∠CAD = 30°
Now, let's suppose that AB = 2
BD =
And, AD2 = AB2 – BD2 = (2
So, AD = √3
Using the definitions of the trigonometric ratios:
Sin 30° =
Cos 30° =
Tan 30° =
Cosec 30° =
Sec 30° =
Cot 30° =
Same for 60°
Sin 60° =
Cos 60° =
Tan 60° = √3
Cosec 60° =
Sec 60° =
Cot 60° =
When the angle is 0°, the perpendicular side becomes 0, and the hypotenuse is equal to the base.
So, in this case,
Sin 0° =
Cos 0° =
Tan 0° =
Cosec 0° =
Sec 0° =
Cot 0° =
When the angle is 90°, the base becomes 0, and the hypotenuse is equal to the perpendicular.
So, in this case,
Sin 90° =
Cos 90° =
Tan 90° =
Cosec 90° =
Sec 90° =
Cot 90° =
Trigonometric identities are fundamental equations that hold for all values of an angle. The three most important identities are:
In ABC, right-angled at B,
AB2 + BC2 = AC2
Dividing each term by AC2,
I.e.
And, from trigonometry ratios, this becomes:
(cos A)2 + (sin A)2 =1
cos2 A + sin2 A = 1
In ABC, right-angled at B,
AB2 + BC2 = AC2
Dividing each term by AB2,
I.e.
And, from trigonometry ratios, this becomes:
1 + (tan A)2 = (sec A)2
1 + tan2 A = sec2 A
In ABC, right-angled at B,
AB2 + BC2 = AC2
Dividing each term by BC2,
I.e.
And, from trigonometry ratios, this becomes:
(cot A)2 + 1 = (cosec A)2
cot2 A + 1 = cosec2 A
Students must download the notes below for each chapter to ace the topics.
Students must check the NCERT Exemplar solutions for class 10 of Mathematics and Science Subjects.
Students must check the NCERT solutions for class 10 of Mathematics and Science Subjects.
To learn about the NCERT books and syllabus, read the following articles and get a direct link to download them.
The word trigonometry is derived from the Greek words ‘tri’ (meaning three), ‘gon’ (meaning sides) and ‘metron’ (meaning measure). Trigonometry is the study of relationships between the sides and angles of a triangle. Mainly, it deals with right-angled triangles, where one angle is always 90°.
The six fundamental trigonometric ratios are:
Sine (sin θ) = Perpendicular / Hypotenuse
Sinθ =
Cosine (cos θ) = Base / Hypotenuse
Cosθ =
Tangent (tan θ) = Perpendicular / Base
Tanθ =
Cosecant (cosec θ) = Hypotenuse / Perpendicular (Reciprocal of sine)
Cosecθ =
Secant (sec θ) = Hypotenuse / Base (Reciprocal of cosine)
Secθ =
Cotangent (cot θ) = Base / Perpendicular (Reciprocal of tangent)
Cotθ =
Use the mnemonic:
"Some People Have Curly Brown Hair, Through Proper Brushing."
For reciprocals:
Trigonometric Ratios:
Sine (sin θ) = Perpendicular / Hypotenuse
Sinθ =
Cosine (cos θ) = Base / Hypotenuse
Cosθ =
Tangent (tan θ) = Perpendicular / Base
Tanθ =
Cosecant (cosec θ) = Hypotenuse / Perpendicular (Reciprocal of sine)
Cosecθ =
Secant (sec θ) = Hypotenuse / Base (Reciprocal of cosine)
Secθ =
Cotangent (cot θ) = Base / Perpendicular (Reciprocal of tangent)
Cotθ =
Trigonometric Identities:
cos2 A + sin2 A = 1
1 + tan2 A = sec2 A
cot2 A + 1 = cosec2 A
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