JEE Exam > JEE Notes > Physics for JEE Main & Advanced > Concept of Angle and T-Ratios
Concept of Angle and T-Ratios | Physics for JEE Main & Advanced PDF Download
| Table of contents |
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| Introduction to Trigonometry |
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| Measurement of Angles |
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| T-ratios (or Trigonometric Functions) |
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| New Definition of T-ratios |
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Introduction to Trigonometry
- The word 'trigonometry' is derived from the Greek words 'trigon' and 'metron' and it means 'measuring the sides of a triangle'. The subject was originally developed to solve geometric problems involving triangles. Sea captains studied it for navigation, surveyors to map out the new lands, by engineers and others.
- Currently, trigonometry is used in many areas such as the science of seismology, designing electric circuits, describing the state of an atom, predicting the heights of tides in the ocean, analysing a musical tone, and many other areas.
Measurement of Angles
There are two systems of measurement of angles:
(i) Sexagesimal or English System: Here 1 right angle = 90 (degrees)
1 = 60’ (minutes)
1’ = 60" (seconds)
(ii) Circular System: Here an angle is measured in radians. One radian corresponds to the angle subtended by an arc of length ’r ’ at the centre of the circle of radius r. It is a constant quantity and does not depend upon the circle's radius.
(a) Relation between the two systems: Degree = Radian*180 / π
(b) If θ is the angle subtended at the centre of a circle of radius 'r',
by an arc of length 'l' then l / r = θ
Representation of Angle in Circular System
Note: l, r are in the same units and θ is always in radians.
Example.1. If the arcs of the same length in two circles subtend angles of 60° and 75° at their centres. Find the ratio of their radii.
Sol: Let r1 and r2 be the radii of the given circles and let their arcs of the same length s subtend angles of 60 and 75 at their centres.
Now,
⇒ 4r1 = 5r2 ⇒ r1 : r2 = 5 : 4
Question for Concept of Angle and T-RatiosTry yourself:Which of the following is correct?
View Solution
T-ratios (or Trigonometric Functions)
A Right Angled Triangle
In a right-angle triangle:
sin θ = p / h
cos θ = b / h
tan θ = p / b
cosec θ = h / p
sec θ = h / b
cot θ = b / p
'p' is perpendicular, 'b' is base and 'h' is hypotenuse.
Note : The quantity by which the cosine falls short of unity i.e. 1 - cos θ, is called the versed sine θ of θ and also by which the sine falls short of unity i.e. 1- sinθ is called the coversed sine of θ.
New Definition of T-ratios
By using rectangular coordinates the definitions of trigonometric functions can be extended to angles of any size in the following way (see diagram). A point P is taken with coordinates (x, y). The radius vector OP has length r and the angle 0 is taken as the directed angle measured anticlockwise from the x-axis.
The three main trigonometric functions are then defined in terms of r and the coordinates x and y:
- sinθ = y / r
- cosθ = x / r
- tanθ = y / x
(The other functions are reciprocals of these)
This can give negative values of the trigonometric functions.Question for Concept of Angle and T-RatiosTry yourself:If tanθ = −4/3 then sinθ is
View Solution
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FAQs on Concept of Angle and T-Ratios - Physics for JEE Main & Advanced
| 1. Can you explain the concept of angles in trigonometry? |  |
Ans. In trigonometry, angles are measured in degrees or radians. A degree is divided into 60 minutes, and a minute is divided into 60 seconds. A radian is a unit of angle, equal to approximately 57.2958 degrees. Angles can be positive or negative depending on their direction of rotation.
| 2. What are T-ratios (Trigonometric Functions) in trigonometry? | |
Ans. T-ratios, also known as trigonometric functions, are ratios of the sides of a right triangle. The main trigonometric functions are sine, cosine, and tangent, denoted as sin, cos, and tan, respectively. These functions relate the angles of a triangle to the lengths of its sides.
| 3. How are T-ratios defined in trigonometry? | |
Ans. In trigonometry, sine (sin), cosine (cos), and tangent (tan) are defined as follows:
- Sine (sin) = Opposite/Hypotenuse
- Cosine (cos) = Adjacent/Hypotenuse
- Tangent (tan) = Opposite/Adjacent
| 4. What is the new definition of T-ratios in trigonometry? | |
Ans. In the new definition of T-ratios, the reciprocal functions of sine, cosine, and tangent are introduced. These functions are cosecant (csc), secant (sec), and cotangent (cot), defined as the reciprocals of sine, cosine, and tangent, respectively.
| 5. How can I apply the concept of angles and T-ratios in JEE exams? | |
Ans. In JEE exams, questions related to angles and T-ratios often involve solving trigonometric equations or finding unknown sides and angles of a triangle using trigonometric functions. It is essential to understand the basic trigonometric concepts and formulas to solve such problems effectively.
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Dec 22, 2024 Last updated |
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