Sum and Product Identities - Trigonometry
Card 1 of 24
Find 
without using a calculator.
Find
Tap to reveal answer
To answer this question, use the sum formula for sine:
To answer this question, use the sum formula for sine:
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Simplify the following expression using trigonometric identities:
Simplify the following expression using trigonometric identities:
Tap to reveal answer
In order to simplify the given equation we should first try to determine if the Pythagorean Theorem as applicable to trigonomety can be utilized. We do this first due to the higher degree of the functions involved. We can notice that if we group the higher order sine and the higher order cosine, that we can in fact pull out some common terms:
Now we notice that we can further group the terms:
The first term in the previous equation is in fact the Pythagorean Theorem as applied to trigonometry and the second term is the sum of two angles with respect to the sine function:
This reduced simply to the sum function for sine:
In order to simplify the given equation we should first try to determine if the Pythagorean Theorem as applicable to trigonomety can be utilized. We do this first due to the higher degree of the functions involved. We can notice that if we group the higher order sine and the higher order cosine, that we can in fact pull out some common terms:
Now we notice that we can further group the terms:
The first term in the previous equation is in fact the Pythagorean Theorem as applied to trigonometry and the second term is the sum of two angles with respect to the sine function:
This reduced simply to the sum function for sine:
← Didn't Know|Knew It →
Without the aide of a calculator, compute the exact value of 
Without the aide of a calculator, compute the exact value of
Tap to reveal answer
In order to determine an exact value we want to split up the angle into angles which we can determine the exact value; these are multiples of 
, 
, 
, 
, and 
. Since the given angle has a 12 in the denominator we must have two fractions which can be found to have a common denominator of 12, the pairs are 3 and 4 or 2 and 6. We can write an algebraic equation which must hold to assist us in determining these values:
Where 
and 
are integers. If we work to get a common denominator of 12:
We can now reduce the equation to obtain a simpler form:
Using this equation it is simple to see that 
and 
and the operation is addition. Therefore rewriting the equation:
Where we have simply reduced the fractions to their lowest terms. Now that we have the sum of two known angles we can simply continue and write out the appropriate formula:
We know the values of the corresponding angles and we must note the quadrant the angle is in, here we are in the first quadrant for both angles which means that all values are positive.
This reduces to:
In order to determine an exact value we want to split up the angle into angles which we can determine the exact value; these are multiples of
Where
We can now reduce the equation to obtain a simpler form:
Using this equation it is simple to see that
Where we have simply reduced the fractions to their lowest terms. Now that we have the sum of two known angles we can simply continue and write out the appropriate formula:
We know the values of the corresponding angles and we must note the quadrant the angle is in, here we are in the first quadrant for both angles which means that all values are positive.
This reduces to:
← Didn't Know|Knew It →
Without the aide of a calculator, compute the exact value of 
Without the aide of a calculator, compute the exact value of
Tap to reveal answer
In order to determine an exact value we want to split up the angle into angles which we can determine the exact value; these are multiples of 
, 
, 
, 
, and 
. Since the given angle has a 12 in the denominator we must have two fractions which can be found to have a common denominator of 12, the pairs are 3 and 4 or 2 and 6. We can write an algebraic equation which must hold to assist us in determining these values:
Where 
and 
are integers. If we work to get a common denominator of 12:
We can now reduce the equation to obtain a simpler form:
Using this equation it is simple to see that 
and 
and the operation is subtraction. Therefore rewriting the equation:
Where we have simply reduced the fractions to their lowest terms. Now that we have the sum of two known angles we can simply continue and write out the appropriate formula:
We know the values of the corresponding angles and we must note the quadrant the angle is in, here we are in the first quadrant for both angles which means that all values are positive.
This reduces to:
In order to determine an exact value we want to split up the angle into angles which we can determine the exact value; these are multiples of
Where
We can now reduce the equation to obtain a simpler form:
Using this equation it is simple to see that
Where we have simply reduced the fractions to their lowest terms. Now that we have the sum of two known angles we can simply continue and write out the appropriate formula:
We know the values of the corresponding angles and we must note the quadrant the angle is in, here we are in the first quadrant for both angles which means that all values are positive.
This reduces to:
← Didn't Know|Knew It →
Without the aide of a calculator, compute the exact value of 
Without the aide of a calculator, compute the exact value of
Tap to reveal answer
In order to determine an exact value we want to split up the angle into angles which we can determine the exact value; these are multiples of 
, 
, 
, 
, and 
. Since the given angle has a 12 in the denominator we must have two fractions which can be found to have a common denominator of 12, the pairs are 3 and 4 or 2 and 6. We can write an algebraic equation which must hold to assist us in determining these values:
Where 
and 
are integers. If we work to get a common denominator of 12:
We can now reduce the equation to obtain a simpler form:
Using this equation we actually need to flip the fractions as follows:
Now we can get the right values of; 
and 
and the operation is subtraction. Therefore rewriting the equation:
Where we have simply reduced the fractions to their lowest terms. Now that we have the sum of two known angles we can simply continue and write out the appropriate formula:
We know the values of the corresponding angles and we must note the quadrant the angle is in, here we are in the first quadrant for angle A which means that all values are positive and the second angle, B, we are in the second quadrant so sine is positive and all other functions are negative.
This reduces to:
Finally, we obtain the solution:
In order to determine an exact value we want to split up the angle into angles which we can determine the exact value; these are multiples of
Where
We can now reduce the equation to obtain a simpler form:
Using this equation we actually need to flip the fractions as follows:
Now we can get the right values of;
Where we have simply reduced the fractions to their lowest terms. Now that we have the sum of two known angles we can simply continue and write out the appropriate formula:
We know the values of the corresponding angles and we must note the quadrant the angle is in, here we are in the first quadrant for angle A which means that all values are positive and the second angle, B, we are in the second quadrant so sine is positive and all other functions are negative.
This reduces to:
Finally, we obtain the solution:
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What is the phase shift of 
?
What is the phase shift of
Tap to reveal answer
The key here is to use this sum/product identity:
In this case, 
and 
. Note as well that because 
subtracts, it will translate into 
. So using the identity, we can state 
as...
Bear in mind that for a sine function of form
...our phase shift is equal to...
The key here is to use this sum/product identity:
In this case,
Bear in mind that for a sine function of form
...our phase shift is equal to...
← Didn't Know|Knew It →
Find without using a calculator.
Find
Tap to reveal answer
To answer this question, use the sum formula for sine:
To answer this question, use the sum formula for sine:
← Didn't Know|Knew It →
Simplify the following expression using trigonometric identities:
Simplify the following expression using trigonometric identities:
Tap to reveal answer
In order to simplify the given equation we should first try to determine if the Pythagorean Theorem as applicable to trigonomety can be utilized. We do this first due to the higher degree of the functions involved. We can notice that if we group the higher order sine and the higher order cosine, that we can in fact pull out some common terms:
Now we notice that we can further group the terms:
The first term in the previous equation is in fact the Pythagorean Theorem as applied to trigonometry and the second term is the sum of two angles with respect to the sine function:
This reduced simply to the sum function for sine:
In order to simplify the given equation we should first try to determine if the Pythagorean Theorem as applicable to trigonomety can be utilized. We do this first due to the higher degree of the functions involved. We can notice that if we group the higher order sine and the higher order cosine, that we can in fact pull out some common terms:
Now we notice that we can further group the terms:
The first term in the previous equation is in fact the Pythagorean Theorem as applied to trigonometry and the second term is the sum of two angles with respect to the sine function:
This reduced simply to the sum function for sine:
← Didn't Know|Knew It →
Without the aide of a calculator, compute the exact value of
Without the aide of a calculator, compute the exact value of
Tap to reveal answer
In order to determine an exact value we want to split up the angle into angles which we can determine the exact value; these are multiples of , , , , and . Since the given angle has a 12 in the denominator we must have two fractions which can be found to have a common denominator of 12, the pairs are 3 and 4 or 2 and 6. We can write an algebraic equation which must hold to assist us in determining these values:
Where and are integers. If we work to get a common denominator of 12:
We can now reduce the equation to obtain a simpler form:
Using this equation it is simple to see that and and the operation is addition. Therefore rewriting the equation:
Where we have simply reduced the fractions to their lowest terms. Now that we have the sum of two known angles we can simply continue and write out the appropriate formula:
We know the values of the corresponding angles and we must note the quadrant the angle is in, here we are in the first quadrant for both angles which means that all values are positive.
This reduces to:
In order to determine an exact value we want to split up the angle into angles which we can determine the exact value; these are multiples of
Where
We can now reduce the equation to obtain a simpler form:
Using this equation it is simple to see that
Where we have simply reduced the fractions to their lowest terms. Now that we have the sum of two known angles we can simply continue and write out the appropriate formula:
We know the values of the corresponding angles and we must note the quadrant the angle is in, here we are in the first quadrant for both angles which means that all values are positive.
This reduces to:
← Didn't Know|Knew It →
Without the aide of a calculator, compute the exact value of
Without the aide of a calculator, compute the exact value of
Tap to reveal answer
In order to determine an exact value we want to split up the angle into angles which we can determine the exact value; these are multiples of , , , , and . Since the given angle has a 12 in the denominator we must have two fractions which can be found to have a common denominator of 12, the pairs are 3 and 4 or 2 and 6. We can write an algebraic equation which must hold to assist us in determining these values:
Where and are integers. If we work to get a common denominator of 12:
We can now reduce the equation to obtain a simpler form:
Using this equation it is simple to see that and and the operation is subtraction. Therefore rewriting the equation:
Where we have simply reduced the fractions to their lowest terms. Now that we have the sum of two known angles we can simply continue and write out the appropriate formula:
We know the values of the corresponding angles and we must note the quadrant the angle is in, here we are in the first quadrant for both angles which means that all values are positive.
This reduces to:
In order to determine an exact value we want to split up the angle into angles which we can determine the exact value; these are multiples of
Where
We can now reduce the equation to obtain a simpler form:
Using this equation it is simple to see that
Where we have simply reduced the fractions to their lowest terms. Now that we have the sum of two known angles we can simply continue and write out the appropriate formula:
We know the values of the corresponding angles and we must note the quadrant the angle is in, here we are in the first quadrant for both angles which means that all values are positive.
This reduces to:
← Didn't Know|Knew It →
Without the aide of a calculator, compute the exact value of
Without the aide of a calculator, compute the exact value of
Tap to reveal answer
In order to determine an exact value we want to split up the angle into angles which we can determine the exact value; these are multiples of , , , , and . Since the given angle has a 12 in the denominator we must have two fractions which can be found to have a common denominator of 12, the pairs are 3 and 4 or 2 and 6. We can write an algebraic equation which must hold to assist us in determining these values:
Where and are integers. If we work to get a common denominator of 12:
We can now reduce the equation to obtain a simpler form:
Using this equation we actually need to flip the fractions as follows:
Now we can get the right values of; and and the operation is subtraction. Therefore rewriting the equation:
Where we have simply reduced the fractions to their lowest terms. Now that we have the sum of two known angles we can simply continue and write out the appropriate formula:
We know the values of the corresponding angles and we must note the quadrant the angle is in, here we are in the first quadrant for angle A which means that all values are positive and the second angle, B, we are in the second quadrant so sine is positive and all other functions are negative.
This reduces to:
Finally, we obtain the solution:
In order to determine an exact value we want to split up the angle into angles which we can determine the exact value; these are multiples of
Where
We can now reduce the equation to obtain a simpler form:
Using this equation we actually need to flip the fractions as follows:
Now we can get the right values of;
Where we have simply reduced the fractions to their lowest terms. Now that we have the sum of two known angles we can simply continue and write out the appropriate formula:
We know the values of the corresponding angles and we must note the quadrant the angle is in, here we are in the first quadrant for angle A which means that all values are positive and the second angle, B, we are in the second quadrant so sine is positive and all other functions are negative.
This reduces to:
Finally, we obtain the solution:
← Didn't Know|Knew It →
What is the phase shift of ?
What is the phase shift of
Tap to reveal answer
The key here is to use this sum/product identity:
In this case, and . Note as well that because subtracts, it will translate into . So using the identity, we can state as...
Bear in mind that for a sine function of form
...our phase shift is equal to...
The key here is to use this sum/product identity:
In this case,
Bear in mind that for a sine function of form
...our phase shift is equal to...
← Didn't Know|Knew It →
Find without using a calculator.
Find
Tap to reveal answer
To answer this question, use the sum formula for sine:
To answer this question, use the sum formula for sine:
← Didn't Know|Knew It →
Simplify the following expression using trigonometric identities:
Simplify the following expression using trigonometric identities:
Tap to reveal answer
In order to simplify the given equation we should first try to determine if the Pythagorean Theorem as applicable to trigonomety can be utilized. We do this first due to the higher degree of the functions involved. We can notice that if we group the higher order sine and the higher order cosine, that we can in fact pull out some common terms:
Now we notice that we can further group the terms:
The first term in the previous equation is in fact the Pythagorean Theorem as applied to trigonometry and the second term is the sum of two angles with respect to the sine function:
This reduced simply to the sum function for sine:
In order to simplify the given equation we should first try to determine if the Pythagorean Theorem as applicable to trigonomety can be utilized. We do this first due to the higher degree of the functions involved. We can notice that if we group the higher order sine and the higher order cosine, that we can in fact pull out some common terms:
Now we notice that we can further group the terms:
The first term in the previous equation is in fact the Pythagorean Theorem as applied to trigonometry and the second term is the sum of two angles with respect to the sine function:
This reduced simply to the sum function for sine:
← Didn't Know|Knew It →
Without the aide of a calculator, compute the exact value of
Without the aide of a calculator, compute the exact value of
Tap to reveal answer
In order to determine an exact value we want to split up the angle into angles which we can determine the exact value; these are multiples of , , , , and . Since the given angle has a 12 in the denominator we must have two fractions which can be found to have a common denominator of 12, the pairs are 3 and 4 or 2 and 6. We can write an algebraic equation which must hold to assist us in determining these values:
Where and are integers. If we work to get a common denominator of 12:
We can now reduce the equation to obtain a simpler form:
Using this equation it is simple to see that and and the operation is addition. Therefore rewriting the equation:
Where we have simply reduced the fractions to their lowest terms. Now that we have the sum of two known angles we can simply continue and write out the appropriate formula:
We know the values of the corresponding angles and we must note the quadrant the angle is in, here we are in the first quadrant for both angles which means that all values are positive.
This reduces to:
In order to determine an exact value we want to split up the angle into angles which we can determine the exact value; these are multiples of
Where
We can now reduce the equation to obtain a simpler form:
Using this equation it is simple to see that
Where we have simply reduced the fractions to their lowest terms. Now that we have the sum of two known angles we can simply continue and write out the appropriate formula:
We know the values of the corresponding angles and we must note the quadrant the angle is in, here we are in the first quadrant for both angles which means that all values are positive.
This reduces to:
← Didn't Know|Knew It →
Without the aide of a calculator, compute the exact value of
Without the aide of a calculator, compute the exact value of
Tap to reveal answer
In order to determine an exact value we want to split up the angle into angles which we can determine the exact value; these are multiples of , , , , and . Since the given angle has a 12 in the denominator we must have two fractions which can be found to have a common denominator of 12, the pairs are 3 and 4 or 2 and 6. We can write an algebraic equation which must hold to assist us in determining these values:
Where and are integers. If we work to get a common denominator of 12:
We can now reduce the equation to obtain a simpler form:
Using this equation it is simple to see that and and the operation is subtraction. Therefore rewriting the equation:
Where we have simply reduced the fractions to their lowest terms. Now that we have the sum of two known angles we can simply continue and write out the appropriate formula:
We know the values of the corresponding angles and we must note the quadrant the angle is in, here we are in the first quadrant for both angles which means that all values are positive.
This reduces to:
In order to determine an exact value we want to split up the angle into angles which we can determine the exact value; these are multiples of
Where
We can now reduce the equation to obtain a simpler form:
Using this equation it is simple to see that
Where we have simply reduced the fractions to their lowest terms. Now that we have the sum of two known angles we can simply continue and write out the appropriate formula:
We know the values of the corresponding angles and we must note the quadrant the angle is in, here we are in the first quadrant for both angles which means that all values are positive.
This reduces to:
← Didn't Know|Knew It →
Without the aide of a calculator, compute the exact value of
Without the aide of a calculator, compute the exact value of
Tap to reveal answer
In order to determine an exact value we want to split up the angle into angles which we can determine the exact value; these are multiples of , , , , and . Since the given angle has a 12 in the denominator we must have two fractions which can be found to have a common denominator of 12, the pairs are 3 and 4 or 2 and 6. We can write an algebraic equation which must hold to assist us in determining these values:
Where and are integers. If we work to get a common denominator of 12:
We can now reduce the equation to obtain a simpler form:
Using this equation we actually need to flip the fractions as follows:
Now we can get the right values of; and and the operation is subtraction. Therefore rewriting the equation:
Where we have simply reduced the fractions to their lowest terms. Now that we have the sum of two known angles we can simply continue and write out the appropriate formula:
We know the values of the corresponding angles and we must note the quadrant the angle is in, here we are in the first quadrant for angle A which means that all values are positive and the second angle, B, we are in the second quadrant so sine is positive and all other functions are negative.
This reduces to:
Finally, we obtain the solution:
In order to determine an exact value we want to split up the angle into angles which we can determine the exact value; these are multiples of
Where
We can now reduce the equation to obtain a simpler form:
Using this equation we actually need to flip the fractions as follows:
Now we can get the right values of;
Where we have simply reduced the fractions to their lowest terms. Now that we have the sum of two known angles we can simply continue and write out the appropriate formula:
We know the values of the corresponding angles and we must note the quadrant the angle is in, here we are in the first quadrant for angle A which means that all values are positive and the second angle, B, we are in the second quadrant so sine is positive and all other functions are negative.
This reduces to:
Finally, we obtain the solution:
← Didn't Know|Knew It →
What is the phase shift of ?
What is the phase shift of
Tap to reveal answer
The key here is to use this sum/product identity:
In this case, and . Note as well that because subtracts, it will translate into . So using the identity, we can state as...
Bear in mind that for a sine function of form
...our phase shift is equal to...
The key here is to use this sum/product identity:
In this case,
Bear in mind that for a sine function of form
...our phase shift is equal to...
← Didn't Know|Knew It →
Find without using a calculator.
Find
Tap to reveal answer
To answer this question, use the sum formula for sine:
To answer this question, use the sum formula for sine:
← Didn't Know|Knew It →
Simplify the following expression using trigonometric identities:
Simplify the following expression using trigonometric identities:
Tap to reveal answer
In order to simplify the given equation we should first try to determine if the Pythagorean Theorem as applicable to trigonomety can be utilized. We do this first due to the higher degree of the functions involved. We can notice that if we group the higher order sine and the higher order cosine, that we can in fact pull out some common terms:
Now we notice that we can further group the terms:
The first term in the previous equation is in fact the Pythagorean Theorem as applied to trigonometry and the second term is the sum of two angles with respect to the sine function:
This reduced simply to the sum function for sine:
In order to simplify the given equation we should first try to determine if the Pythagorean Theorem as applicable to trigonomety can be utilized. We do this first due to the higher degree of the functions involved. We can notice that if we group the higher order sine and the higher order cosine, that we can in fact pull out some common terms:
Now we notice that we can further group the terms:
The first term in the previous equation is in fact the Pythagorean Theorem as applied to trigonometry and the second term is the sum of two angles with respect to the sine function:
This reduced simply to the sum function for sine:
← Didn't Know|Knew It →