Fundamental Trigonometric Identities - Pre-Calculus
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State
in terms of sine and cosine.
State in terms of sine and cosine.
The definition of tangent is sine divided by cosine.

The definition of tangent is sine divided by cosine.
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Which of the following statements is false?
Which of the following statements is false?
Of the six trigonometric functions, four are odd, meaning
. These four are:
- sin x
- tan x
- cot x
- csc x
That leaves two functions which are even, which means that
. These are:
- cos x
- sec x
Of the aforementioned, only
is incorrect, since secant is an even function, which implies that 
Of the six trigonometric functions, four are odd, meaning . These four are:
- sin x
- tan x
- cot x
- csc x
That leaves two functions which are even, which means that . These are:
- cos x
- sec x
Of the aforementioned, only is incorrect, since secant is an even function, which implies that
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Find the exact value
.
Find the exact value
.
By the double angle formula

![sin\left[2tan^{-1}\left(\frac{3}{4}\right)\right]=2sin\left[tan^{-1}\left(\frac{3}{4}\right)\right]cos\left[tan^{-1}\left(\frac{3}{4}\right)\right]](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/278625/gif.latex)

By the double angle formula
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Find the exact value
.
Find the exact value
.
By the double angle formula

![sin\left[2tan^{-1}\left(\frac{8}{15}\right)\right]=2sin\left[tan^{-1}\left(\frac{8}{15}\right)\right]cos\left[tan^{-1}\left(\frac{8}{15}\right)\right]](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/278713/gif.latex)

By the double angle formula
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Find the exact value
.
Find the exact value
.
By the double-angula formula for cosine

For this problem
![cos\left[2tan^{-1}\left(\frac{5}{12}\right)\right]=cos^2\left[tan^{-1}\left(\frac{5}{12}\right)\right]-sin^2\left[tan^{-1}\left(\frac{5}{12}\right)\right]](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/278721/gif.latex)

By the double-angula formula for cosine
For this problem
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Compute
in function of
.
Compute in function of
.
Using trigonometric identities we have :
and we know that:

This gives us :


Hence:

Using trigonometric identities we have :
and we know that:
This gives us :
Hence:
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Find the exact value
.
Find the exact value
.
By the double-angle formula for the sine function

we have

thus the double angle formula becomes,
![sin\left[2sin^{-1}\left(\frac{12}{37}\right)\right]=2sin\left[sin^{-1}\left(12/37\right)\right]cos\left[sin^{-1}\left(12/37\right)\right]](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/293362/gif.latex)

By the double-angle formula for the sine function
we have
thus the double angle formula becomes,
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Which of the following is equivalent to 
Which of the following is equivalent to
When trying to identify equivalent equations that use trigonometric functions it is important to recall the general formula and understand how the terms affect the translations.
The general formula for sine is as follows.
where
is the amplitude,
is used to find the period of the function
,
represents the phase shift
, and
is the vertical shift.
This is also true for,
.
Looking at the possible answer choices lets first focus on the ones containing sine.
has a vertical shift of
therefore it is not an equivalent function as it is moving the original function up.
has a phase shift of
therefore it is not an equivalent function as it is moving the original function to the right.
Now lets shift our focus to the answer choices that contain cosine.
has a vertical shift down of
units. This will create a graph that has a range that is below the
-axis. It is important to remember that
has a range of
. Therefore this cosine function is not an equivalent equation.
has a phase shift to the right
units. Plugging in some values we see that,
,
.
Now, looking back at our original function and plugging in those same values of
and
we get,
,
.
Since the function values are the same for each of the input values, we can conclude that
is equivalent to
.
When trying to identify equivalent equations that use trigonometric functions it is important to recall the general formula and understand how the terms affect the translations.
The general formula for sine is as follows.
where
is the amplitude,
is used to find the period of the function
,
represents the phase shift
, and
is the vertical shift.
This is also true for,
.
Looking at the possible answer choices lets first focus on the ones containing sine.
has a vertical shift of
therefore it is not an equivalent function as it is moving the original function up.
has a phase shift of
therefore it is not an equivalent function as it is moving the original function to the right.
Now lets shift our focus to the answer choices that contain cosine.
has a vertical shift down of
units. This will create a graph that has a range that is below the
-axis. It is important to remember that
has a range of
. Therefore this cosine function is not an equivalent equation.
has a phase shift to the right
units. Plugging in some values we see that,
,
.
Now, looking back at our original function and plugging in those same values of and
we get,
,
.
Since the function values are the same for each of the input values, we can conclude that is equivalent to
.
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If
, which of the following best represents
?
If , which of the following best represents
?
The expression
is a double angle identity that can also be rewritten as:

Replace the value of theta for
.
The correct answer is: 
The expression is a double angle identity that can also be rewritten as:
Replace the value of theta for .
The correct answer is:
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Compute

Compute
A useful trigonometric identity to remember is

If we plug in
into this equation, we get

We can divide the equation by 2 to get

A useful trigonometric identity to remember is
If we plug in into this equation, we get
We can divide the equation by 2 to get
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Compute

Compute
A useful trigonometric identity to remember for this problem is

or equivalently,

If we substitute
for
, we get

A useful trigonometric identity to remember for this problem is
or equivalently,
If we substitute for
, we get
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Using the half-angle identities, which of the following answers best resembles
?
Using the half-angle identities, which of the following answers best resembles ?
Write the half angle identity for sine.

Since we are given
, the angle is equal to
. Set these two angles equal to each other and solve for
.


Substitute this value into the formula.

Write the half angle identity for sine.
Since we are given , the angle is equal to
. Set these two angles equal to each other and solve for
.
Substitute this value into the formula.
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Let
and
two reals. Given that:


What is the value of:
?
Let and
two reals. Given that:
What is the value of:
?
We have:


and :


(1)-(2) gives:
![4sin(a)cos(b)cos(a)sin(b)=[2sin(a)cos(a)] [2sin(b)cos(b)]$$](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/333489/gif.latex)
Knowing from the above formula that:( take a=b in the formula above)

This gives:

We have:
and :
(1)-(2) gives:
Knowing from the above formula that:( take a=b in the formula above)
This gives:
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Let
,
, and
be real numbers. Given that:


What is the value of
in function of
?
Let ,
, and
be real numbers. Given that:
What is the value of in function of
?
We note first, using trigonometric identities that:



This gives:

Since, 
We have :

We note first, using trigonometric identities that:
This gives:
Since,
We have :
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Using the fact that,
.
What is the result of the following sum:

Using the fact that,
.
What is the result of the following sum:
We can write the above sum as :


From the given fact, we have :




and we have :
.
This gives :

We can write the above sum as :
From the given fact, we have :
and we have : .
This gives :
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Given that :

Let,

What is
in function of
?
Given that :
Let,
What is in function of
?
We will use the given formula :
We have in this case:

Since we know that :

This gives :

We will use the given formula :
We have in this case:
Since we know that :
This gives :
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Using the fact that
, what is the result of the following sum:

Using the fact that , what is the result of the following sum:
We can write the above sum as :

From the given fact, we have :




This gives us :


Therefore we have:

We can write the above sum as :
From the given fact, we have :
This gives us :
Therefore we have:
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Let
be real numbers. If
and 
What is the value of
in function of
?
Let be real numbers. If
and
What is the value of in function of
?
Using trigonometric identities we know that :

This gives :

We also know that 
This gives :

Using trigonometric identities we know that :
This gives :
We also know that
This gives :
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Given that :
and,

Compute :
in function of
.
Given that :
and,
Compute :
in function of
.
We have using the given result:



This gives us:


Hence :

We have using the given result:
This gives us:
Hence :
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Let
be an integer and
a real number. Compute
as a function of
.
Let be an integer and
a real number. Compute
as a function of
.
Using trigonometric identities we have :

We know that :
and 
This gives :

Using trigonometric identities we have :
We know that :
and
This gives :
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