TRIGONOMETRIC FUNCTIONS

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]

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

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

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

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

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)

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

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

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)

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).

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

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

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

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.

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

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

sin^2 ϴ + cos^2 ϴ = 1

cos(-ϴ) = cos ϴ

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

 cos(α+β) = cos α cos β - sin α sin β

cos(α-β) = cos α cos β + sin α sin β

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

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

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 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]

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}

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

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

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

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.

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).

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

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

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)

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

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

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

sec( α/2) = 1/( √(1-cos α)/2)

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

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

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

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)

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).

sin^2ϴ + cos^2ϴ = 1

cotϴ = 1/tanϴ

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

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

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

cot(α/2) =± 1/√(1-cosα)/(1+cosα)

cot(2α) = (1-tan^2(α))/(2tan2α)

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

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

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

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

tanϴ = sinϴ/cosϴ

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

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

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

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

tan(α/2) =± √(1-cosα)/(1+cosα)

tan(2α) =(2tan2α)/(1-tan^2(α))

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}