by siby sebastian pgt(maths)

siby sebastian pgt maths

TRIGONOMETRIC FUNCTIONSBASIC IDEAS

BY SIBY SEBASTIAN PGT(MATHS)


The rotation of the terminal side of an angle

counterclockwise.

The rotation of the terminal side is clockwise.

AC

B

Basic Terms


The most common unit for measuring angles is the degree. (One rotation = 360o)

¼ rotation = 90o, ½ rotation = 180o,

Angle and measure of angle are not the same, but it is common to say that an angle = its measure

Types of angles named on basis of measure:

Angle Measures and Types of Angles


So far we have measured angles in degrees

For most practical applications of trigonometry this is the preferred

measure

For advanced mathematics courses it is more common to measure angles in units called

“radian measure”

Measuring Angles


An angle with its vertex at the center of a circle of radius ‘r’ units subtended by an arc of length ‘r’ unit is 1 radian. (1 rad)

Radian Measure


Since a complete rotation of a ray

back to the initial position generates

a circle of radius “r”, and the

circumference of that circle (arc

length) is 2, there are 2 radians in a

complete rotation

Based on the reasoning just

discussed:2rad = 3600 ,

rad = 1800

1 rad =

Comments on Radian Measure


Multiply a degree measure by

and simplify to convert to radians.

Multiply a radian measure by and simplify to convert to degrees.

Conversion Between Degrees and Radians


a) 60

b) 221.7 221.70 =

Convert Degrees to Radians


a) = x =

b) 3.25 rad3.25 rad = x

Convert Radians to Degrees


Equivalent Angles in Degrees and Radians


Trigonometric FunctionsIn a circle of radius ‘r’ units and if P(x,y) is a point on the circle then the trigonometric functions are defined bysin cosec

cos sec

tan cot =

x

yr


Trigonometric Functions

“Circular Functions” are named as trig

functions (sine, cosine, tangent, etc.)

The domain of trig functions is a

set of angles measured either

in degrees or radians

The domain of circular functions is the set of real

numbers

The value of a trig function of a specific angle in its domain is a ratio of real

numbers

The value of circular

function of a real number

“x” is the same as the

corresponding trig function of

“x radians”


• sin2 A = (sin A)2

• tan3A = (tanA)3

• Sec5A = (secA)5

Exponential Notation and Trigonometric Functions


• Considering the following three functions and the sign of x, y and r in each quadrant, which functions are positive in each quadrant?

Signs of Trig Functions by Quadrant of Angle


It will help to memorize by learning these words in

Quadrants I - IV:“All students take calculus”And remembering reciprocal

identitiesTrig functions are negative in quadrants where they are not

positive

Mnemonic Techniques


Given an angle A in standard position, and (x,y) a point on the terminal side a distance of r > 0 from the origin, sin A = y/r

Domain of sine function is the set of all A for which y/r is a real number. Since r can’t be zero, y/r is always a real number and domain is “any angle”

Range of sine function is the set of all y/r, but since y is less than or equal to r, this ratio will always be equal to 1 or will be a proper fraction, positive or negative:

Domain and Range of Sine Function


GRAPH OF sine FUNCTION

Click here to see how sin function is generated

http://www.mathopenref.com/triggraphsine.html


Given an angle A in standard position, and (x,y) a point on the terminal side a distance of r > 0 from the origin, cos A = x/r

Domain of cosine function is the set of all A for which x/r is a real number. Since r can’t be zero, x/r is always a real number and domain is “any angle”

Range of cosine function is the set of all x/r, but since x is less than or equal to r, this ratio will always be equal to 1, -1 or will be a proper fraction, positive or negative:

Domain and Range of Cosine Function


GRAPH OF cosine FUNCTION

Click here to see how cosine function is generated

http://www.mathopenref.com/triggraphcosine.html


Given an angle A in standard position, and (x,y) a point on the terminal side a distance of r > 0 from the origin, tan A = y/x

Domain of tangent function is the set of all A for which y/x is a real number. Tangent will be undefined when x = 0, therefore domain is all angles except for odd multiples of 90o

Range of tangent function is the set of all y/x, but since all of these are possible: x=y, x<y, x>y, this ratio can be any positive or negative real number:

Domain and Range of Tangent Function


GRAPH OF tangent FUNCTION

Click here to see how tangent function is generated

http://www.mathopenref.com/triggraphtan.html


Given an angle A in standard position, and (x,y) a point on the terminal side a distance of r > 0 from the origin, csc A = r/y

Domain of cosecant function is the set of all A for which r/y is a real number. Cosecant will be undefined when y = 0, therefore domain is all angles except for integer multiples of 180o

Range of cosecant function is the reciprocal of the range of the sine function. Reciprocals of numbers between -1 and 1 are:

Domain and Range of Cosecant Function


GRAPH OF cosecant FUNCTION


Given an angle A in standard position, and (x,y) a point on the terminal side a distance of r > 0 from the origin, sec A = r/x

Domain of secant function is the set of all A for which r/x is a real number. Secant will be undefined when x = 0, therefore domain is all angles except for odd multiples of 90o

Range of secant function is the reciprocal of the range of the cosine function. Reciprocals of numbers between -1 and 1 are:

Domain and Range of Secant Function


GRAPH OF secant FUNCTION


Given an angle A in standard position, and (x,y) a point on the terminal side a distance of r > 0 from the origin, cot A = x/y

Domain of cotangent function is the set of all A for which x/y is a real number. Cotangent will be undefined when y = 0, therefore domain is all angles except for integer multiples of 180o

Range of cotangent function is the reciprocal of the range of the tangent function. The reciprocal of the set of numbers between negative infinity and positive infinity is:

Domain and Range of Cotangent Function


GRAPH OF cotangent FUNCTION


1 sin 1 1 cos 1tan and cot can take any real number

sec 1 or sec 1

csc 1 or csc 1.

Ranges of Trigonometric FunctionsFor any angle for which the indicated functions exist:

Note that sec and csc are never between 1 and 1


Periodic Properties


Theorem Even-Odd Properties


BASIC RULES OF TRIGONOMETRIC FUNCTIONS

1.sin(

2.cos() = sinx

3.tan( = cotx



4.sin(

5.cos() = - sinx

6.tan( = - cotx



7.sin(

8.cos() = -cosx

9.tan( = - tanx



10.sin(

11.cos() = -cosx

12.tan( = tanx



13.sin(

14.cos() = -sinx

15.tan( = cotx



16. sin( 17 .cos() = sinx18 .tan( = - cotx



19.sin(

20.cos() = cosx

21.tan(2 =-tanx



1. sin2x +cos2x =1

2. 1+tan2x =sec2x

3. 1+cot2x =cosec2x



SUM AND DIFFERENCE OF TWO ANGLES

1.cos(x + y) = cosxcosy – sinxsiny2.cos(x – y) = cosxcosy + sinxsiny3.sin(x + y) = sinxcosy + cosxsiny4.sin( x – y) = sinxcosy - cosxsiny



5.tan(x + y) = 6.tan(x – y) = 7.cot(x + y) = 8.cot(x - y) =



PRODUCT AS SUM OR DIFFERENCE1 .2sinxcosy = sin(x + y) + sin(x – y)2. 2cosxsiny = sin(x + y) – sin(x – y)3.2cosxcosy = cos(x + y)+cos(x – y)4.-2sinxsiny = cos(x + y) – cos(x – y)



SUM OR DIFFERENCE AS PRODUCT1.sinx + siny = 2sin(2.sinx – siny = 2cos(3.cosx + cosy = 2cos(4.cosx – cosy = - 2sin(



MULTIPLE ANGLES1.sin2x = 2sinxcosx =

2.cos2x = cos2x – sin2x = 2cos2x – 1 = 1 – 2sin2x =



3.tan2x =

4.sin3x = 3sinx – 4sin3x

5.cos3x = 4cos3x – 3cosx

6.tan3x =



SUB MULTIPLE ANGLES

1.sinx = 2sin2.cosx =

3.1- cosx = 24.1+cosx = 2



GENERAL SOLUTIONS

1.sinx =0 then x= n, n

2.cosx = 0 then x=(2n + 1) n

3.tanx =0 then x= n, n



4.Sinx = siny then,x = n

5.cosx =cosy then

6.tanx = tany then x= n



Sine Rule



Cosine RulecosA = cosB = cosC =


Finally let us dance together and enjoy trigonometry

PracticePractice&PracticeUntil you get it. ……..

by siby sebastian pgt(maths)

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