Identities with Angle Sums - Trigonometry
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Find the exact value of the expression:

Find the exact value of the expression:
There are two ways to solve this problem. If one recognizes the identity
,
the answer is as simple as:

If one misses the identity, or wishes to be more thorough, you can simplify:


There are two ways to solve this problem. If one recognizes the identity
,
the answer is as simple as:
If one misses the identity, or wishes to be more thorough, you can simplify:
Compare your answer with the correct one above
Given
, what is
?
Given , what is
?
We need to use the formula

Substituting
, and
,

We need to use the formula
Substituting , and
,
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Find the exact value of
using
.
Find the exact value of using
.
Our basic sum formula for cosine is:

Substituting the relevant angles gives us:

Now substitute in the exact values for each function, simplifying to keep radicals out of the denominator:

Multiply and subtract to obtain:

Our basic sum formula for cosine is:
Substituting the relevant angles gives us:
Now substitute in the exact values for each function, simplifying to keep radicals out of the denominator:
Multiply and subtract to obtain:
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Find the exact value of the expression:

Find the exact value of the expression:
The formula for the cosine of the difference of two angles is

Substituting, we find that

and

Therefore, what we are really looking for is

Thus,

The formula for the cosine of the difference of two angles is
Substituting, we find that
and
Therefore, what we are really looking for is
Thus,
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Find the exact value of
using
and
.
Find the exact value of using
and
.
The sum identity for tangent states that

Substituting known values for
and
, we have

For ease, multiply all terms by
to get
.
At this point, multiply both halves of the fraction by the conjugate of the denominator:

Finally, simplify.

So,
.
The sum identity for tangent states that
Substituting known values for and
, we have
For ease, multiply all terms by to get
.
At this point, multiply both halves of the fraction by the conjugate of the denominator:
Finally, simplify.
So, .
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Suppose we have two angles,
and
, such that:


Furthermore, suppose that angle
is located in the first quadrant and angle
is located in the fourth.
What is the measure of:

Suppose we have two angles, and
, such that:
Furthermore, suppose that angle is located in the first quadrant and angle
is located in the fourth.
What is the measure of:
We can calculate some missing values using the pythagorean identities.


(Note the negative sign, because
is in the fourth quadrant, where the sine of the angle is always negative).

Note the positive value, since
is in the first quadrant, where cosine is positive.
Now using the rules for double angles:


And then the angle subtraction formula:



We can calculate some missing values using the pythagorean identities.
(Note the negative sign, because is in the fourth quadrant, where the sine of the angle is always negative).
Note the positive value, since is in the first quadrant, where cosine is positive.
Now using the rules for double angles:
And then the angle subtraction formula:
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Calculate
.
Calculate .
Recall the formula for the sine of the sum of two angles:

Here, we can evaluate
by noticing that
and applying the above formula to the sines and cosines of these two angles.



Hence,

Recall the formula for the sine of the sum of two angles:
Here, we can evaluate by noticing that
and applying the above formula to the sines and cosines of these two angles.
Hence,
Compare your answer with the correct one above
What is the value of
, using the sum formula.
What is the value of , using the sum formula.
The formula for
.
We can expand
,
where
and
.
Substituting these values into the equation, we get
.
The final answer is -1, using what we know about the unit circle values.
The formula for
.
We can expand
,
where and
.
Substituting these values into the equation, we get
.
The final answer is -1, using what we know about the unit circle values.
Compare your answer with the correct one above
Simplify the given expression.

Simplify the given expression.
This problem requires the use of two angle sum/difference identities:


Using these identities, we get

which simplifies to

which equals

This problem requires the use of two angle sum/difference identities:
Using these identities, we get
which simplifies to
which equals
Compare your answer with the correct one above
Compare your answer with the correct one above
Find the exact value of the expression:

Find the exact value of the expression:
There are two ways to solve this problem. If one recognizes the identity
,
the answer is as simple as:

If one misses the identity, or wishes to be more thorough, you can simplify:


There are two ways to solve this problem. If one recognizes the identity
,
the answer is as simple as:
If one misses the identity, or wishes to be more thorough, you can simplify:
Compare your answer with the correct one above
Given
, what is
?
Given , what is
?
We need to use the formula

Substituting
, and
,

We need to use the formula
Substituting , and
,
Compare your answer with the correct one above
Find the exact value of
using
.
Find the exact value of using
.
Our basic sum formula for cosine is:

Substituting the relevant angles gives us:

Now substitute in the exact values for each function, simplifying to keep radicals out of the denominator:

Multiply and subtract to obtain:

Our basic sum formula for cosine is:
Substituting the relevant angles gives us:
Now substitute in the exact values for each function, simplifying to keep radicals out of the denominator:
Multiply and subtract to obtain:
Compare your answer with the correct one above
Find the exact value of the expression:

Find the exact value of the expression:
The formula for the cosine of the difference of two angles is

Substituting, we find that

and

Therefore, what we are really looking for is

Thus,

The formula for the cosine of the difference of two angles is
Substituting, we find that
and
Therefore, what we are really looking for is
Thus,
Compare your answer with the correct one above
Find the exact value of
using
and
.
Find the exact value of using
and
.
The sum identity for tangent states that

Substituting known values for
and
, we have

For ease, multiply all terms by
to get
.
At this point, multiply both halves of the fraction by the conjugate of the denominator:

Finally, simplify.

So,
.
The sum identity for tangent states that
Substituting known values for and
, we have
For ease, multiply all terms by to get
.
At this point, multiply both halves of the fraction by the conjugate of the denominator:
Finally, simplify.
So, .
Compare your answer with the correct one above
Suppose we have two angles,
and
, such that:


Furthermore, suppose that angle
is located in the first quadrant and angle
is located in the fourth.
What is the measure of:

Suppose we have two angles, and
, such that:
Furthermore, suppose that angle is located in the first quadrant and angle
is located in the fourth.
What is the measure of:
We can calculate some missing values using the pythagorean identities.


(Note the negative sign, because
is in the fourth quadrant, where the sine of the angle is always negative).

Note the positive value, since
is in the first quadrant, where cosine is positive.
Now using the rules for double angles:


And then the angle subtraction formula:



We can calculate some missing values using the pythagorean identities.
(Note the negative sign, because is in the fourth quadrant, where the sine of the angle is always negative).
Note the positive value, since is in the first quadrant, where cosine is positive.
Now using the rules for double angles:
And then the angle subtraction formula:
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Calculate
.
Calculate .
Recall the formula for the sine of the sum of two angles:

Here, we can evaluate
by noticing that
and applying the above formula to the sines and cosines of these two angles.



Hence,

Recall the formula for the sine of the sum of two angles:
Here, we can evaluate by noticing that
and applying the above formula to the sines and cosines of these two angles.
Hence,
Compare your answer with the correct one above
What is the value of
, using the sum formula.
What is the value of , using the sum formula.
The formula for
.
We can expand
,
where
and
.
Substituting these values into the equation, we get
.
The final answer is -1, using what we know about the unit circle values.
The formula for
.
We can expand
,
where and
.
Substituting these values into the equation, we get
.
The final answer is -1, using what we know about the unit circle values.
Compare your answer with the correct one above
Simplify the given expression.

Simplify the given expression.
This problem requires the use of two angle sum/difference identities:


Using these identities, we get

which simplifies to

which equals

This problem requires the use of two angle sum/difference identities:
Using these identities, we get
which simplifies to
which equals
Compare your answer with the correct one above
Compare your answer with the correct one above