# Class 10 – Trigonometry

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Trigonometry
• Measurements of angle finding trigonometric ratios
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## Study Material

 Introduction: The measure of an angle is the amount of rotation from the direction of one ray of the angle to the other. The initial and final positions of the revolving ray are respectively called the initial and terminal sides (arms). Also, the revolving line is called the generating side, eg, if initial and final positions of the revolving ray are OP and OQ, then the angle formed will be ∠POQ. If the rotation is in clockwise sense, the angle measured is negative and if the rotation is in anti-clockwise sense, the angle measured is positive. System of measurement of angles: There are three system of measurement of an angle. i) Sexagesimal system ii) Centesimal system iii) Circular system i)               Sexagesimal system:In this system one right angle is divided into 90 equal parts and each part is known as a degree. Thus a right angle is equal to 90 degrees. One degree is denoted as 1°. Each degree is divided into 60 equal parts each of which is known as one minute. One minute is denoted as . Each minute consist of 60 parts, each part is known as a second. One second is denoted by . Hence, 1 right angle = (90 degrees)             1 degree =                  1 minute =   ii)             Centesimal system: In this system one right angle is divided into 100 equal parts and each part is known as a grade. Thus one right angle is equal to 100 grades. One grade is denoted as 1g. One grade is divided into 100 equal parts, each part is known as a minute and is denoted as . One minute is also divided into 100 equal parts, each part is known as a second which is denoted by . Hence, 1 right angle = 1 grade =     1 minute =

iii)            Circular system: If the angle subtended by an arc of length l to the centre of circle of radius ‘r’, is θthen θ= .

If the length of arc is equal to the radius of the circle, then the angle subtended at the centre of the circle will be one radian. One radian is denoted by 1C.

The ratio of the circumference of the circle to the diameter of the circle is denoted by a greek letter π and it is a constant quantity.

This constant π  is an irrational quantity and its approximate value is  and its general value upto six places of decimal is 3.142857 (3.14).

Relation between three systems of measurements of an angle:

Let D be the number of degrees, ‘C’R be the number of radians and ‘G’ be the number of grades in an angle θ.

This is the required relation between the three systems of measurement of an angle.

If measure of an angle is given in degree, then to convert it into radians, multiply the measure of an angle by  and if the measure of an angle is given in radians, then to convert it into degrees, write  at the place of .

Example: If the three angles of a quadrilateral are  radians. Find the fourth angle.

Solution: First angle =

Second angle =

Third angle =  =

Fourth angle =

Trigonometric ratios: Let us take a right angled triangle ABC right angled at B.

Let CAB = θ, then

Trigonometric ratios of any angle is always constant.

Fundamental relation among trigonometric ratios: It is clear from the definitions of trigonometric ratios that

i)

ii)

iii)

iv)

v)

Trigonometric ratios of 0°: Consider a right angle triangle ABC right angled at C. Now A is made smaller and smaller in the right ΔABC till it becomes zero. If we decrease the A then the side BC also decrease and point B gets closer to the point C and finally when A becomes 0, the point B will coincide to C. AB becomes almost equal to AC.

So, AB = AC and BC = 0

i)

ii)

iii)

iv)

v)

vi)

Trigonometric ratios of 30° and 60°:

Consider an equilateral ΔABC in which the side is of length 2a. Each angle of ΔABC is of 60°. Draw AD perpendicular toon BC.

Since ABC is an equilateral triangle.

AD is the bisector of A and D is the mid point of BC.

Thus in  is a right angled triangle, ,
where

By using Pythagoras theorem,

In ∆ ABD,

I.

II.

Trigonometric ratios of 45°:

Consider a isosceles right triangle ABC with right angle at B such that,

Hence,

Let. Then by Pythagoras theorem.

By Pythagoras theorem

In

Trigonometric ratio of 90°: Consider a right angle triangle ABC right angled at C. A of ΔABC is made large and large till it becomes 90°. If a point A move towards C, then A increases and side AC decreases and closer to point C finally A becomes to 90°. The point A will coincide to C, AB almost equal to BC. So AB = BC and AC = 0.

 Ratio /θ 0° 30° 45° 60° 90° sin θ 0 1 cos θ 1 0 tan θ 0 1 Not defined cosec θ Not defined 2 1 sec θ 1 2 Not defined cot θ Not defined 1 0

Sign of trigonometric ratios in different quadrants:

 Quadrants I II III IV sin, cosec + + – – cos, sec + – – + tan, cot + – + –

If the angles are expressed in degrees then

i) In case of allied angles ,  remains the same.

ii) In case of allied angles then trigonometric ratios changed as.

Trigonometric Identities: An equation involving trigonometric ratios of an angle is called a trigonometric identityies, it isif it is true for all values of the angles (s) involved.

i)

ii)

iii)

From above identities

i)

ii)

iii)

.

Solution: and

Now,

Hence proved.