Class 9 – Trigonometry

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Trigonometry
  • Measure of an angle
  • Trigonometric ratios and identities
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Trigonometry
  • Elimination of an angle
  • Allied angles
  • Compound angles
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Introduction: We begin our study of angles by learning what they are, how to name them, and ways in which angles can be classified. These concepts are explained in earlier classes. Now we learn how to measure an angle, which units can be used to find measure of an angle below.

Angle: The motion of any revolving line in a plane from its initial position (initial side)
to the final position (terminal side) is called angle. The end point O about which the

 line rotates is called the vertex of the angle.

 

Measure of an angle: The measure of an angle is the amount of rotation from the
initial side to the terminal side.

Sense of an angle: The sense of an angle is determined by the direction of rotation of
the initial side to the terminal side.

i)     The sign of an angle is said to be positives, if the rotation of the terminal side is anti-clock wise direction..

ii)   The sign an angle is said to be negative, if the rotation of the terminal side is clockwise direction.

 

Quadrants: Let and  be two mutually perpendicular lines in any plance. These lines divide the plane into four regions called quadrants, numbered counter anti-clockwise. Thus the quadrants  are respectively called the first quadrant, the second quadrant, the third quadrant and the fourth quadrant.

θ will be more than 360° after making one complete revolution, the revolving line does not stop at the initial position OX but proceeds  further on.

 

Measuring angles: There are three systems for measuring an angle.

i)     Sexagesimal or English system

ii)   Centesimal or French system

iii)  Circular system

i)     Sexagesimal or English system: Here a right angle is divided into 90 equal parts known as degrees. Each degree is divided into 60 equal parts called minutes and each minute is further divided into 60 equal parts called seconds. Therefore,
1 right angle = 90 degree (90°)


ii)   Centesimal or French system: Here a right angle is divided into 100 equal parts called grades and each grade is divided into 100 equal parts, called minutes and each minute is further divided into 100 seconds. Therefore,
1 right angle = 100 grades (=100 g)

iii)  Circular system: The unit of measurement of angle in this system is a radian (or 1C). A radian is defined as the angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle and it is denoted by 1C.

In the above figure,  (radius of the circle) and  radian or 1c.

Since the whole circle subtends an angle of 3600 (= 4 right angles) at the centre and the angles at the centre of a circle are in the ratio of subtending arcs, therefore,

This means that a radian is a constant angle, independent of the radius of the circle. Also, we find that

 

Example: The following angles in radian measure and centesimal measure:

i) 45°            ii) 20°         iii) 50°

Solution:

 

Example: One angle of a triangle is 54° and another angle is radian. Find the third angle in centesimal measure.

Solution: Let
B = 45°

Thus,


N     
Now, since 90° =  grade   

 Third angle

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Conversion of degree to radians and vice – versa.
                                           

 

Relationship 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.

Now 90°=1 right angle  right angle

 right angle  right angle            …………….(i)

again,  right angle  right angle

 right angle  right angle  …………….(ii)

and 100g = 1 right angle right angle

 right angle  right angle       …………….(iii)

From (i), (ii) and (iii) we get

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

 

Relationship between an Arc and an Angle: If l is the length of an arc of a circle of radius r, then the angle = or  i.e., Length of arc = radius  angle in radians.

Proof: Let radians be the angle subtended by an arc AP, l be the length of arc AP of a circle with the center O. Cut of an arc of length AB = radius (r) and join OB, then

Now

(Angles at the centre of a circle are proportional to the arcs on which they stand.)

 

Example: An arc AB of a circle subtends an angle xcradians at the centre O of the circle. Given that area of the sector AOB is equal to square of the length of the arc AB, find the value of x.

Solution: Let arc AB = l

Given that area of the sector

AOB = (length of arc AB)2

         

 

Example: The circular wire of radius 7 cm is cut and bend again into an arc of a circle of radius 12 cm. Find the angle subtended by an arc at the centre of the circle.

Solution: Given the diameter of a circular wire = 14 cm.

Therefore length of wire = 14cm

Hence, required angle=radian

Example: The degree measure corresponding to  the given radian

Solution: We have, radian=180°

         

 

Example: The angles of a quadrilateral are in A.P. and the greatest angle is 120°, then find the angles of quadrilateral in radians.

Solution: Let the angles in degrees be

Sum of the angles
= 90°

Also greatest angle=
Hence,

Hence the angles are 90°-30°, 90°-10°, 90°+10° and 90°+30°

Hence the angles are 60°, 80°, 100° and 120°

In terms of radians, the angles are  and  next line that i

 Angles of a quadrilateral are .

Example: The minute hand of a clock is 14 0cm long. How far does the tip of the hand move in 20 minutes?

Solution: We know that the tip of the minute hand makes one complete round in one
hour i.e. 60 minutes since the length of the hand is 14 0cm

The distance moved by the tip in 60 minutes=

Hence the distance in 20 minutes =

Here r = 14 cm

Angle subtended by minute hand in 20 min

 

Important tips: (In a clock):

i)     The angle subtended by hour hand in 12 hours is 360°

The angle subtended by hour hand in 1 hour is 30°

The angle subtended by hour hand in 1 minute  is

ii)   The angle subtended by minute hand in 1 hour is 360°

 The angle subtended by minute hand in 1 minute is 6°