TRIGONOMETRIC FUNCTIONS

COSINE

PROPERTIES OF COSINE

DOMAINE AND RANGE

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THE DOMAINE OF THE COSINE FUNCTION IS THE SET OF ALL REAL NUMBERS, COS(X) • Domain: IRTHE RANGE OF THE COSINE FUNCTION CONSISTS OF ALL REAL NUMBERS BETWEEN -1AND 1, INCLUSIVE • Range: [−1,1]

SINGS OF COS

QUADRANT I

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BETWEEN THE INTERVAL [0,90] DEGREE,COSINE OF ANY ANGLE IN THAT INTERVAL IS POSITIVE,COS(X)>0

QUADRANT II

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BETWEEN THE INTERVAL [90,180] DEGREE,COSINE OF ANY ANGLE IN THAT INTERVAL IS NEGATIVE,COS(X)<0

QUADRANT III

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BETWEEN THE INTERVAL [180,270] DEGREE,COSINE OF ANY ANGLE IN THAT INTERVAL IS NEGATIVE,COS(X)<0

QUADRANT IV

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BETWEEN THE INTERVAL [270,0] DEGREE,COSINE OF ANY ANGLE IN THAT INTERVAL IS POSITIVE,COS(X)>0

PERIOD OF COSINE

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DEFINITIONA function f is called periodic if there is a positive number p such that, whenever ϴ is in the domain of f, so is ϴ+pf(ϴ+p)= f(ϴ). ϴ is the argument of the function.the cosine function is periodic with period 2π      cos(ϴ+2π)=cos(ϴ)

EVEN ODD PROPERTY

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A function f is even if f(-ϴ)=f(ϴ) for all ϴ in the domain of f, function f is odd if f(-ϴ)=-f(ϴ)for all ϴ in the domain of f. cos(-ϴ)=cos(ϴ)proof see book chapter 6 section 6.3 page 389

GRAPH OF COSINE

GRAPH

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Since we want to graph cosine in the xy-plane,we shall use the traditional symbol x for the independent variable(or argument) and y for the dependent variable(or value at x).so we write y=f(x)=cos(x). The independent variable x represents an angle measured in radians.since the cosine function has period 2π,we only need to graph y=f(x)=cos(x) on the interval [0,2π],the remainder will consist of repetitions of this portion.graph see book chapter 6 section 6.4 figure 48 , table 7 page 395

AMPLITUDE

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In general, the value of the function y = A cos(x), where A≠0, will always satisfy the inequality -|A| ≤ Acos(x) ≤ |A|. The number |A| is called the amplitude of y = A cos(x)

PERIODICITY

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The periodicity is given by T=2π/w. We know that the graph of y = cos(wx) is obtained from the graph of y = cos(x) by performing a horizontal compression or stretch by a factor of 1/w. This horizontal compression replaces the interval[0,2π], which contains one period of the graph of y = cos(x), by the interval [0,2π/w], which contains one period of the graph of y = cos(wx).

INVERSE COSINE

DOMAINE

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THE COSINE FUNCTION IS NOT A ONE-TO-ONE FUNCTION WHEN WE CONSIDER ITS ENTIRE DOMAIN, HOWEVER, WE CAN RESTRICT THAT DOMAIN TO GET AN INVERSE FUNCTION. If we restrict the domain of y = cos(x) to the interval [0,π], the restricted function y = cos(x)   0 ≤ x ≤ π is one-to-one and hence have an inverse function which will be obtained by interchanging x and y in the function y =f(x) = cos(x) . The implicit form of the inverse is x = cos(y) , 0 ≤ y ≤ π and we obtain y = cos^-1(x) where -1 ≤ x ≤ 1 and  0 ≤ y ≤ π

RANGE

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The range of the inverse cosine function is the restricted domain [0,π]

PROPERTIES

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For the cosine function and its inverse, the followings properties hold:f^-1(f(x)) = cos^-1(cos(x)) = x ,0 ≤ x ≤ πf(f^-1(x)) = cos(cos^-1(x))=x, -1 ≤ x ≤ 1

IDENTITIES

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Two functions f and g are said to be identically equal if               f(x) = g(x)for every value of x for which both functions are defined. Such an equation is referred to as an identity.

QUOTIENT IDENTITY

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tan ϴ = sin ϴ/cos ϴ hence cos ϴ = sin ϴ/tan ϴ And cot ϴ = cos ϴ / sin ϴ hence      cos ϴ = cot ϴ/ sin ϴ

RECIPROCICAL IDENTITY

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sec ϴ cos ϴ = 1 hence cos ϴ = 1 / sec ϴ

PYTHAGORIAN IDENTITY

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sin^2 ϴ + cos^2 ϴ = 1

EVEN ODD IDENTITY

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cos(-ϴ) = cos ϴ

DERIVATE IDENTITIES

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Use sum and difference to establish identities cos(π/2 -ϴ) = sinϴuse double angle formula to establish identitycos^2 ϴ =(1+cos 2ϴ)/2

FORMULAS FOR COSINE

SUM

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 cos(α+β) = cos α cos β - sin α sin β

DIFFERENCE

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cos(α-β) = cos α cos β + sin α sin β

DOUBLE ANGLE

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Double angle formulascos(2ϴ) = cos^2 ϴ - sin^2 ϴcos(2ϴ) = 1- 2sin^2 ϴcos(2ϴ) = 2cos^2 ϴ - 1

HALF ANGLE

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Half angle formulas for cosinecos α/2 = √(1-cos α)/2

PRODUCT TO SUM

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product to sum formulas cos α cos β  = (1/2)[cos(α-β)+cos(α+β)]sin α cos β = (1/2)[sin(α+β)+sin(α-β)] 

SUM TO PRODUCT

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Sum to Product formulascos α + cos β = 2[cos(α+β)/2 cos(α-β)/2]cos α - cos β =-2[sin (α+β)/2 sin(α-β)/2]

SECANT

PROPERTIE OF SECANT

DOMAIN AND RANGE

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THE DOMAINE OF THE SECANT FUNCTION IS THE SET OF ALL REAL NUMBERS EXCEPT ODD INTEGERS MULTIPLES OF π/2(90 degrees) • Domain: IR/{X≠Kπ/2},K is an odd integerTHE RANGE OF THE SECANT FUNCTION CONSISTS OF ALL REAL NUMBERS LESS THAN OR EQUAL TO -1 OR GREATER THAN OR EQUAL TO 1. THAT IS • Range: IR/{Y≤-1 OR Y≥1}

SIGNS OF SECANT

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BETWEEN THE INTERVAL [0,90] DEGREE,SECANTE OF ANY ANGLE IN THAT INTERVAL IS POSITIVE,SEC(X)>0BETWEEN THE INTERVAL [90,180] DEGREE,SECANTE OF ANY ANGLE IN THAT INTERVAL IS NEGATIVE,SEC(X)<0BETWEEN THE INTERVAL [180,270] DEGREE,SECANTE OF ANY ANGLE IN THAT INTERVAL IS NEGATIVE,SEC(X)<0BETWEEN THE INTERVAL [270,0] DEGREE,SECANTE OF ANY ANGLE IN THAT INTERVAL IS POSITIVE,SEC(X)>0

PERIODICITY

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THE SECANTE FUNCTION LIKE THE COSINE FUNCTION IS A PERIODIC FUNCTION WITH PERIOD 2πSEC(ϴ+2π)=SEC(ϴ)

EVEN ODD PROPERTIES

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THE SECANTE FUNCTION IS AN EVEN FUNCTION WHICH MEANS:SEC(-ϴ)=SEC(ϴ)

GRAPH OF SECANT

GRAPH

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Since we want to graph secant in the xy-plane,we shall use the traditional symbol x for the independent variable(or argument) and y for the dependent variable(or value at x).so we write y=f(x)=sec(x). The independent variable x represents an angle measured in radians.since the cosine function has period 2π,we only need to graph y=f(x)=sec(x) on the interval [0,2π],the remainder will consist of repetitions of this portion.

AMPLITUDE AND
PERIODICITY

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In general, the value of the function y = A sec(x), where A≠0, will always satisfy the inequality -|A| ≤ Acos(x) ≤ |A|. The number |A| is called the amplitude of y = A sec(x)The periodicity is given by T=2π/w. We know that the graph of y = cos(wx) is obtained from the graph of y = cos(x) by performing a horizontal compression or stretch by a factor of 1/w. This horizontal compression replaces the interval[0,2π], which contains one period of the graph of y = cos(x), by the interval [0,2π/w], which contains one period of the graph of y = cos(wx).

IDENTITIES

RECIPROCAL
IDENTITY

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sec ϴ cos ϴ = 1 hence sec ϴ = 1 / cos ϴ

PYTHAGORIAN
IDENTITY

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sin^2 ϴ + cos^2 ϴ = 1divide both side by cos^2 ϴ ,we will havetan^2 ϴ + 1 = sec^2 ϴ

DERIVATE IDENTITIES

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Use sum and difference to establish identities cos(π/2 -ϴ) = sinϴ1/cos(π/2 -ϴ) = 1/sinϴsec(π/2 -ϴ) = cscϴuse double angle formula to establish identitycos^2 ϴ =(1+cos 2ϴ)/2sec^2 ϴ = 2/(1+cos 2ϴ)

INVERSE OF
SECANT

DOMAIN AND
RANGE

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THE SECANT FUNCTION IS NOT A ONE-TO-ONE FUNCTION WHEN WE CONSIDER ITS ENTIRE DOMAIN, HOWEVER, WE CAN RESTRICT THAT DOMAIN TO GET AN INVERSE FUNCTION. If we restrict the domain of y = sec(x) to the interval [0,π/2), the restricted function y = sec(x)  , 0 ≤ x < π/2 is one-to-one and hence have an inverse function which will be obtained by interchanging x and y in the function y =f(x) = sec(x) . The implicit form of the inverse is x = sec(y) , 0 ≤ y < π/2 and we obtain y = sec^-1(x) where 0 ≤ y < π/2

PROPERTIES

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FOR THE SECANT AND ITS INVERSE FUNCTION, THE FOLLOWING PROPERTIES HOL:SEC^-1(SEC X) = X ; [0,pi/2)SEC(SEC^-1 X) = X ; X LESS THAN or equal to -1 or X greater than or equal to 1

FORMULAS FOR
SECANT

DOUBLE ANGLE

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sec(2ϴ) =1/(cos^2 ϴ - sin^2 ϴ)sec(2ϴ) = 1/(1- 2sin^2 ϴ)sec(2ϴ) = 1/( 2cos^2 ϴ - 1)

HALF ANGLE

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sec( α/2) = 1/( √(1-cos α)/2)

SINE

PROPERTIES OF
SINE

DOMAIN AND
RANGE

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The domain of the sine function is the set of all real numbers.The range is the set of all real numbers between -1 and 1 inclusive

SIGNS OF SINE

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The sine function is :-positive in the first and second quadrant;-negative in the third and fourth quadrant

EVEN ODD PROPERTIE

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The sine function is an odd function there for ,sin(-ϴ) = -sinϴ

GRAPH OF SINE

GRAPH

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In general, the value of the function y = A sin(x), where A≠0, will always satisfy the inequality -|A| ≤ A sin(x) ≤ |A|. The number |A| is called the amplitude of y = A sin(x)

AMPLITUDE AND
PERIODICITY

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The periodicity is given by T=2π/w. We know that the graph of y = sin(wx) is obtained from the graph of y = sin(x) by performing a horizontal compression or stretch by a factor of 1/w. This horizontal compression replaces the interval[0,2π], which contains one period of the graph of y = sin(x), by the interval [0,2π/w], which contains one period of the graph of y = sin(wx).

INVERSE OF SINE

DOMAIN AND
RANGE

PROPERTIES

IDENTITIES

QUOTIENT IDENTITY

RECIPROCAL
IDENTITY

PYTHAGORIAN
IDENTITY

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sin^2ϴ + cos^2ϴ = 1

DERIVATE IDENTITY

FORMULAS OF
SINE

SUM AND
DIFFERENCE

DOUBLE
ANGLE

HALF
ANGLE

PRODUCT
TO SUM

SUM TO
PRODUCT

COTANGENTE

PROPERTIES OF
COTANGENTE

DOMAIN AND
RANGE

SIGNS OF
COTANGENTE

EVEN ODD
PROPERTY

GRAPH OF
COTANGENTE

GRAPH

AMPLITUDE AND
PERIODICITY

INVERSE OF
COTANGENTE

DOMAIN AND
RANGE

PROPERTIES

IDENTITIES

QUOTIENT

RECIPROCAL

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cotϴ = 1/tanϴ

PYTHAGORIAN

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using the pythagorian identitycos^2ϴ +sin^2ϴ = 1 , and by dividing both sides by sin^ϴ, we can obtain:cot^2ϴ +1 = csc^2ϴ 

EVEN ODD

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COTANGENT IS AN ODD FUNCTION LIKE TANGENT;cot(-ϴ) = -cot(ϴ)

FORMULAS OF
COTANGENTE

SUM AND
DIFFERENCE

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cot(α+β) = (1-tanαtanβ)/(tanα+tanβ)cot(α-β) = (1+tanαtanβ)/(tanα-tanβ)

DOUBLE ANGLE

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cot(α/2) =± 1/√(1-cosα)/(1+cosα)

HALPH ANGLE

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cot(2α) = (1-tan^2(α))/(2tan2α)

PRODUCT TO
SUM

SUM TO
PRODUCT

TANGENTE

PROPERTIES OF
TANGENT

DOMAIN AND
RANGE

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The domain of tangent function is the set of all real numbers except odd integers multiple of kΠ

SIGNS

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The tangent function is: -positive in the first and third quadrant -negative in the second and fourth quadrant

EVEN ODD
PROPERTY

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the tangent function is an odd function;tan(-ϴ) = -tan ϴ

GRAPH OF
TANGENT

GRAPH

AMPLITUDE AND
PERIODICITY

INVERSE OF
TANGENT

DOMAIN AND
RANGE

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The domain of the inverse function of tangent is negative infinity to infinity and the range from zero inclusive 90 degree exclusive

PROPERTIES

IDENTITIES

QUOTIENT

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tanϴ = sinϴ/cosϴ

RECIPROCAL

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Reciprocal Identitytan ϴ = sinϴ/cosϴ and cotϴ = cosϴ/sinϴhence tanϴ = 1/cotϴ

PYTHAGORIAN

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by using the pythagorian theorem we can demonstrate that:tan^2(ϴ) +1 = sec^2(ϴ)

EVEN ODD

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The tangent function is an odd function;tan(-ϴ) = tan ϴ

FORMULAS OF
TANGENT

SUM AND
DIFFERENCE

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tan(α+β) = (tanα+tanβ)/(1-tanαtanβ)tan(α-β) = (tanα-tanβ)/(1+tanαtanβ)

DOUBLE ANGLE

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tan(α/2) =± √(1-cosα)/(1+cosα)

HALF ANGLE

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tan(2α) =(2tan2α)/(1-tan^2(α))

COSECANTE

PROPERTIES OF
COSECANT

DOMAIN AND
RANGE

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THE DOMAINE OF THE COSECANT FUNCTION IS THE SET OF ALL REAL NUMBERS EXCEPT  INTEGERS MULTIPLES OF π(180 degrees) • Domain: IR/{X≠Kπ},K is an  integerTHE RANGE OF THE COSECANT FUNCTION CONSISTS OF ALL REAL NUMBERS LESS THAN OR EQUAL TO -1 OR GREATER THAN OR EQUAL TO 1. THAT IS • Range: IR/{Y≤-1 OR Y≥1}

SIGNS OF
COSECANT

EVEN ODD
PROPERTIES

GRAPH OF
COSECANT

GRAPH

AMPLITUDE AND
PERIODICITY

INVERSE OF
COSECANT

DOMAIN AND
RANGE

PROPERTIES

IDENTITIES

RECIPROCAL
IDENTITY

PYTHAGORIAN
IDENTITY

EVEN ODD
IDENTITY

FORMULAS OF
COSECANT

DOUBLE ANGLE

HALF ANGLE