Trigonometric Functions - Trigonometry
Card 0 of 20
For which values of 
, where 
in the unit circle, is 
undefined?
For which values of
Recall that 
. Since the ratio of any two real numbers is undefined when the denominator is equal to 
, 
must be undefined for those values of 
where 
. Restricting our attention to those values of 
between 
and 
, 
when 
or 
. Hence, 
is undefined when 
or 
.
Recall that
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What is the domain of f(x) = sin x?
What is the domain of f(x) = sin x?
The domain of a function is the range of all possible inputs, or x-values, that yield a real value for f(x). Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Sine is defined as the ratio between the side length opposite to the angle in question and the hypotenuse (SOH, or sin x = opposite/hypotenuse). In any triangle created by the angle x and the x-axis, the hypotenuse is a nonzero number. As a result, the denominator of the fraction created by the definition sin x = opposite/hypotenuse is not equal to zero for any angle value x. Therefore, the domain of f(x) = sin x is all real numbers.
The domain of a function is the range of all possible inputs, or x-values, that yield a real value for f(x). Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Sine is defined as the ratio between the side length opposite to the angle in question and the hypotenuse (SOH, or sin x = opposite/hypotenuse). In any triangle created by the angle x and the x-axis, the hypotenuse is a nonzero number. As a result, the denominator of the fraction created by the definition sin x = opposite/hypotenuse is not equal to zero for any angle value x. Therefore, the domain of f(x) = sin x is all real numbers.
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What is the domain of f(x) = cos x?
What is the domain of f(x) = cos x?
The domain of a function is the range of all possible inputs, or x-values, that yield a real value for f(x). Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Cosine is defined as the ratio between the side length opposite to the angle in question and the hypotenuse (CAH, or cos x = adjacent/hypotenuse). In any triangle created by the angle x and the x-axis, the hypotenuse is a nonzero number. As a result, the denominator of the fraction created by the definition cos x = adjacent/hypotenuse is not equal to zero for any angle value x. Therefore, the domain of f(x) = cos x is all real numbers.
The domain of a function is the range of all possible inputs, or x-values, that yield a real value for f(x). Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Cosine is defined as the ratio between the side length opposite to the angle in question and the hypotenuse (CAH, or cos x = adjacent/hypotenuse). In any triangle created by the angle x and the x-axis, the hypotenuse is a nonzero number. As a result, the denominator of the fraction created by the definition cos x = adjacent/hypotenuse is not equal to zero for any angle value x. Therefore, the domain of f(x) = cos x is all real numbers.
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Which of the following trigonometric functions is undefined?
Which of the following trigonometric functions is undefined?
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Secant is the reciprocal of cosine, so the secant of any angle x for which cos x = 0 must be undefined, since it would have a denominator equal to 0. The value of cos (pi/2) is 0, so the secant of (pi)/2 must be undefined.
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Secant is the reciprocal of cosine, so the secant of any angle x for which cos x = 0 must be undefined, since it would have a denominator equal to 0. The value of cos (pi/2) is 0, so the secant of (pi)/2 must be undefined.
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Which of the following trigonometric functions is undefined?
Which of the following trigonometric functions is undefined?
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Cotangent is the reciprocal of tangent, so the cotangent of any angle x for which tan x = 0 must be undefined, since it would have a denominator equal to 0. The value of tan (pi) is 0, so the cotangent of (pi) must be undefined.
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Cotangent is the reciprocal of tangent, so the cotangent of any angle x for which tan x = 0 must be undefined, since it would have a denominator equal to 0. The value of tan (pi) is 0, so the cotangent of (pi) must be undefined.
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Which of the following trigonometric functions is undefined?
Which of the following trigonometric functions is undefined?
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Secant is the reciprocal of cosine, so the secant of any angle x for which cos x = 0 must be undefined, since it would have a denominator equal to 0. The value of cos 3(pi/2) is 0, so the secant of 3(pi)/2 must be undefined.
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Secant is the reciprocal of cosine, so the secant of any angle x for which cos x = 0 must be undefined, since it would have a denominator equal to 0. The value of cos 3(pi/2) is 0, so the secant of 3(pi)/2 must be undefined.
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Which of the following trigonometric functions is undefined?
Which of the following trigonometric functions is undefined?
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Cotangent is the reciprocal of tangent, so the cotangent of any angle x for which tan x = 0 must be undefined, since it would have a denominator equal to 0. The value of tan (0) is 0, so the cotangent of (0) must be undefined.
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Cotangent is the reciprocal of tangent, so the cotangent of any angle x for which tan x = 0 must be undefined, since it would have a denominator equal to 0. The value of tan (0) is 0, so the cotangent of (0) must be undefined.
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Which of the following trigonometric functions is undefined?
Which of the following trigonometric functions is undefined?
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Cosecant is the reciprocal of sine, so the cosecant of any angle x for which sin x = 0 must be undefined, since it would have a denominator equal to 0. The value of sin (0) is 0, so the cosecant of 0 must be undefined.
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Cosecant is the reciprocal of sine, so the cosecant of any angle x for which sin x = 0 must be undefined, since it would have a denominator equal to 0. The value of sin (0) is 0, so the cosecant of 0 must be undefined.
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Which of the following trigonometric functions is undefined?
Which of the following trigonometric functions is undefined?
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Tangent is defined as the ratio between the side length opposite to the angle in question and the side length adjacent to it (TOA, or tan x = opposite/adjacent). In a triangle created by the angle x and the x-axis, the adjacent side length lies along the x-axis; however, when the angle x lies on the y-axis, no length can be drawn along the x-axis to represent the angle. As a result, the denominator of the fraction created by the definition tan x = opposite/adjacent is equal to zero for any angle along the y-axis (90 or 270 degrees, or pi/2 or 3pi/2 in radians.) Therefore, tan 3(pi)/2 is undefined.
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Tangent is defined as the ratio between the side length opposite to the angle in question and the side length adjacent to it (TOA, or tan x = opposite/adjacent). In a triangle created by the angle x and the x-axis, the adjacent side length lies along the x-axis; however, when the angle x lies on the y-axis, no length can be drawn along the x-axis to represent the angle. As a result, the denominator of the fraction created by the definition tan x = opposite/adjacent is equal to zero for any angle along the y-axis (90 or 270 degrees, or pi/2 or 3pi/2 in radians.) Therefore, tan 3(pi)/2 is undefined.
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Which of the following trigonometric functions is undefined?
Which of the following trigonometric functions is undefined?
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Cosecant is the reciprocal of sine, so the cosecant of any angle x for which sin x = 0 must be undefined, since it would have a denominator equal to 0. The value of sin (pi) is 0, so the cosecant of pi must be undefined.
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Cosecant is the reciprocal of sine, so the cosecant of any angle x for which sin x = 0 must be undefined, since it would have a denominator equal to 0. The value of sin (pi) is 0, so the cosecant of pi must be undefined.
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Which of the following trigonometric functions is undefined?
Which of the following trigonometric functions is undefined?
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Tangent is defined as the ratio between the side length opposite to the angle in question and the side length adjacent to it (TOA, or tan x = opposite/adjacent). In a triangle created by the angle x and the x-axis, the adjacent side length lies along the x-axis; however, when the angle x lies on the y-axis, no length can be drawn along the x-axis to represent the angle. As a result, the denominator of the fraction created by the definition tan x = opposite/adjacent is equal to zero for any angle along the y-axis (90 or 270 degrees, or pi/2 or 3pi/2 in radians.) Therefore, tan (pi)/2 is undefined.
Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Tangent is defined as the ratio between the side length opposite to the angle in question and the side length adjacent to it (TOA, or tan x = opposite/adjacent). In a triangle created by the angle x and the x-axis, the adjacent side length lies along the x-axis; however, when the angle x lies on the y-axis, no length can be drawn along the x-axis to represent the angle. As a result, the denominator of the fraction created by the definition tan x = opposite/adjacent is equal to zero for any angle along the y-axis (90 or 270 degrees, or pi/2 or 3pi/2 in radians.) Therefore, tan (pi)/2 is undefined.
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True or False: The inverse of the function 
is also a function.
True or False: The inverse of the function 
Consider the graph of the function 
. It passes the vertical line test, that is if a vertical line is drawn anywhere on the graph it only passes through a single point of the function. This means that 
is a function.
Now, for its inverse to also be a function it must pass the horizontal line test. This means that if a horizontal line is drawn anywhere on the graph it will only pass through one point.
This is not true, and we can also see that if we graph the inverse of 
(
) that this does not pass the vertical line test and therefore is not a function. If you wish to graph the inverse of 
, then you must restrict the domain so that your graph will pass the vertical line test.
Consider the graph of the function 
Now, for its inverse to also be a function it must pass the horizontal line test. This means that if a horizontal line is drawn anywhere on the graph it will only pass through one point.
This is not true, and we can also see that if we graph the inverse of 
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Which of the following is the graph of the inverse of 
with 
?
Which of the following is the graph of the inverse of 
Note that the inverse of 
is not 
, that is the reciprocal. The inverse of 
is 
also written as 
. The graph of 
with 
is as follows.
And so the inverse of this graph must be the following with 
and 
Note that the inverse of 
And so the inverse of this graph must be the following with 
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Which best describes the easiest method to graph an inverse trigonometric function (or any function) based on the parent function?
Which best describes the easiest method to graph an inverse trigonometric function (or any function) based on the parent function?
To find an inverse function you swap the 
and 
values. Take 
for example, to find the inverse we use the following method.
(swap the 
and 
values)
(solving for 
)
To find an inverse function you swap the 
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Which of the following represents the graph of 
with 
?
Which of the following represents the graph of 
If we are looking for the graph of 
with 
, that means this is the inverse of 
with 
. The graph of 
with 
is
Switching the 
and 
values to graph the inverse we get the graph
If we are looking for the graph of 
Switching the 
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Which of the following is the graph of 
with 
?
Which of the following is the graph of 
We first need to think about the graph of the function 
.
Using the formula 
where 
is the vertical shift, we have to perform a transformation of moving the function 
up two units on the graph.
We first need to think about the graph of the function 
Using the formula 
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Which of the following is the correct graph and range of the inverse function of 
with 
?
Which of the following is the correct graph and range of the inverse function of 
First, we must solve for the inverse of 
So now we are trying to find the range of and plot the function 
. Let’s start with the graph of 
. We know the domain is 
.
Now using the formula 
where 
= Period, the period of 
is 
. And so we perform a transformation to the graph of 
to change the period from 
to 
.
We can see that the graph has a range of 
First, we must solve for the inverse of
So now we are trying to find the range of and plot the function 
Now using the formula 
We can see that the graph has a range of
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True or False: The domain for 
will always be all real numbers no matter the value of 
or any transformations applied to the tangent function.
True or False: The domain for 
This is true because just as the range of 
is all real numbers due to the vertical asymptotes of the function, the function 
extends to all values of 
but is limited in its values of 
. No matter the transformations applied, all values of 
will still be reached.
This is true because just as the range of 
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Which of the following is the graph of 
?
Which of the following is the graph of 
First, we must consider the graph of 
.
Using the formula 
we can apply the transformations step-by-step. First we will transform the amplitude, so 
so we must shorten the amplitude to 
.
Now we must apply a vertical shift of one unit since 
. This leaves us with our answer.
First, we must consider the graph of 
Using the formula 
Now we must apply a vertical shift of one unit since 
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Simplify the following trionometric function:
Simplify the following trionometric function:
To solve the problem, you need to know the following information:
Replace the trigonometric functions with these values:
To solve the problem, you need to know the following information:
Replace the trigonometric functions with these values:
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