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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 integer
THE 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}
tan(2α) =(2tan2α)/(1-tan^2(α))
tan(α/2) =± √(1-cosα)/(1+cosα)
tan(α+β) = (tanα+tanβ)/(1-tanαtanβ)
tan(α-β) = (tanα-tanβ)/(1+tanαtanβ)
The tangent function is an odd function;
tan(-ϴ) = tan ϴ
by using the pythagorian theorem we can demonstrate that:
tan^2(ϴ) +1 = sec^2(ϴ)
Reciprocal Identity
tan ϴ = sinϴ/cosϴ and cotϴ = cosϴ/sinϴ
hence tanϴ = 1/cotϴ
tanϴ = sinϴ/cosϴ
The domain of the inverse function of tangent is negative infinity to infinity and the range from zero inclusive 90 degree exclusive
the tangent function is an odd function;
tan(-ϴ) = -tan ϴ
The tangent function is:
-positive in the first and third quadrant
-negative in the second and fourth quadrant
The domain of tangent function is the set of all real numbers except odd integers multiple of kΠ
cot(2α) = (1-tan^2(α))/(2tan2α)
cot(α/2) =± 1/√(1-cosα)/(1+cosα)
cot(α+β) = (1-tanαtanβ)/(tanα+tanβ)
cot(α-β) = (1+tanαtanβ)/(tanα-tanβ)
COTANGENT IS AN ODD FUNCTION LIKE TANGENT;
cot(-ϴ) = -cot(ϴ)
using the pythagorian identity
cos^2ϴ +sin^2ϴ = 1 , and by dividing both sides by sin^ϴ, we can obtain:
cot^2ϴ +1 = csc^2ϴ
cotϴ = 1/tanϴ
sin^2ϴ + cos^2ϴ = 1
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).
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 sine function is an odd function there for ,
sin(-ϴ) = -sinϴ
The sine function is :
-positive in the first and second quadrant;
-negative in the third and fourth quadrant
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
sec( α/2) = 1/( √(1-cos α)/2)
sec(2ϴ) =1/(cos^2 ϴ - sin^2 ϴ)
sec(2ϴ) = 1/(1- 2sin^2 ϴ)
sec(2ϴ) = 1/( 2cos^2 ϴ - 1)
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
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
Use sum and difference to establish identities
cos(π/2 -ϴ) = sinϴ
1/cos(π/2 -ϴ) = 1/sinϴ
sec(π/2 -ϴ) = cscϴ
use double angle formula to establish identity
cos^2 ϴ =(1+cos 2ϴ)/2
sec^2 ϴ = 2/(1+cos 2ϴ)
sin^2 ϴ + cos^2 ϴ = 1
divide both side by cos^2 ϴ ,we will have
tan^2 ϴ + 1 = sec^2 ϴ
sec ϴ cos ϴ = 1 hence sec ϴ = 1 / cos ϴ
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).
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.
THE SECANTE FUNCTION IS AN EVEN FUNCTION WHICH MEANS:
SEC(-ϴ)=SEC(ϴ)
THE SECANTE FUNCTION LIKE THE COSINE FUNCTION IS A PERIODIC FUNCTION WITH PERIOD 2π
SEC(ϴ+2π)=SEC(ϴ)
BETWEEN THE INTERVAL [0,90] DEGREE,
SECANTE OF ANY ANGLE IN THAT INTERVAL IS POSITIVE,
SEC(X)>0
BETWEEN THE INTERVAL [90,180] DEGREE,
SECANTE OF ANY ANGLE IN THAT INTERVAL IS NEGATIVE,
SEC(X)<0
BETWEEN THE INTERVAL [180,270] DEGREE,
SECANTE OF ANY ANGLE IN THAT INTERVAL IS NEGATIVE,
SEC(X)<0
BETWEEN THE INTERVAL [270,0] DEGREE,
SECANTE OF ANY ANGLE IN THAT INTERVAL IS POSITIVE,
SEC(X)>0
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 integer
THE 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}
Sum to Product formulas
cos 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]
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)]
Half angle formulas for cosine
cos α/2 = √(1-cos α)/2
Double angle formulas
cos(2ϴ) = cos^2 ϴ - sin^2 ϴ
cos(2ϴ) = 1- 2sin^2 ϴ
cos(2ϴ) = 2cos^2 ϴ - 1
cos(α-β) = cos α cos β + sin α sin β
cos(α+β) = cos α cos β - sin α sin β
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.
Use sum and difference to establish identities
cos(π/2 -ϴ) = sinϴ
use double angle formula to establish identity
cos^2 ϴ =(1+cos 2ϴ)/2
cos(-ϴ) = cos ϴ
sin^2 ϴ + cos^2 ϴ = 1
sec ϴ cos ϴ = 1 hence cos ϴ = 1 / sec ϴ
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
For the cosine function and its inverse, the followings properties hold:
f^-1(f(x)) = cos^-1(cos(x)) = x ,0 ≤ x ≤ π
f(f^-1(x)) = cos(cos^-1(x))=x, -1 ≤ x ≤ 1
The range of the inverse cosine function is the restricted domain [0,π]
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 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).
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)
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
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
DEFINITION
A 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+p
f(ϴ+p)= f(ϴ). ϴ is the argument of the function.
the cosine function is periodic with period 2π
cos(ϴ+2π)=cos(ϴ)
QUADRANT IV
BETWEEN THE INTERVAL [270,0] DEGREE,
COSINE OF ANY ANGLE IN THAT INTERVAL IS POSITIVE,
COS(X)>0
QUADRANT III
BETWEEN THE INTERVAL [180,270] DEGREE,
COSINE OF ANY ANGLE IN THAT INTERVAL IS NEGATIVE,
COS(X)<0
QUADRANT II
BETWEEN THE INTERVAL [90,180] DEGREE,
COSINE OF ANY ANGLE IN THAT INTERVAL IS NEGATIVE,
COS(X)<0
QUADRANT I
BETWEEN THE INTERVAL [0,90] DEGREE,
COSINE OF ANY ANGLE IN THAT INTERVAL IS POSITIVE,
COS(X)>0
THE DOMAINE OF THE COSINE FUNCTION IS THE SET OF ALL REAL NUMBERS, COS(X)
• Domain: IR
THE RANGE OF THE COSINE FUNCTION CONSISTS OF ALL REAL NUMBERS BETWEEN -1AND 1, INCLUSIVE
• Range: [−1,1]
