Trigonometry : Sum and Product Identities

Study concepts, example questions & explanations for Trigonometry

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Example Questions

Example Question #1 : Sum And Product Identities

Find  without using a calculator.

Possible Answers:

Correct answer:

Explanation:

To answer this question, use the sum formula for sine:

Example Question #2 : Sum And Product Identities

Simplify the following expression using trigonometric identities:

Possible Answers:

Can not be further reduced

Correct answer:

Explanation:

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:

Example Question #3 : Sum And Product Identities

Without the aide of a calculator, compute the exact value of 

Possible Answers:

Correct answer:

Explanation:

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:

 

Example Question #4 : Sum And Product Identities

Without the aide of a calculator, compute the exact value of 

Possible Answers:

Correct answer:

Explanation:

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:

 

Example Question #5 : Sum And Product Identities

Without the aide of a calculator, compute the exact value of 

Possible Answers:

Correct answer:

Explanation:

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:

 

Example Question #5 : Sum And Product Identities

What is the phase shift of  ?

Possible Answers:

Correct answer:

Explanation:

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...

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