Trigonometry : Sum, Difference, and Product Identities

Study concepts, example questions & explanations for Trigonometry

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Example Questions

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Example Question #1 : Complete A Proof Using Sums, Differences, Or Products Of Sines And Cosines

True or false:

.

Possible Answers:

Cannot be determined

False

True 

Correct answer:

False

Explanation:

The sum of sines is given by the formula .

Example Question #2 : Complete A Proof Using Sums, Differences, Or Products Of Sines And Cosines

True or false: .

Possible Answers:

True

False

Cannot be determined

Correct answer:

False

Explanation:

The difference of sines is given by the formula .

Example Question #3 : Complete A Proof Using Sums, Differences, Or Products Of Sines And Cosines

True or false: .

Possible Answers:

True

False

Cannot be determined

Correct answer:

False

Explanation:

The sum of cosines is given by the formula .

Example Question #4 : Complete A Proof Using Sums, Differences, Or Products Of Sines And Cosines

True or false: .

Possible Answers:

False

True

Cannot be determined

Correct answer:

False

Explanation:

The difference of cosines is given by the formula .

Example Question #5 : Complete A Proof Using Sums, Differences, Or Products Of Sines And Cosines

Which of the following correctly demonstrates the compound angle formula?

Possible Answers:

Correct answer:

Explanation:

The compound angle formula for sines states that .

Example Question #6 : Complete A Proof Using Sums, Differences, Or Products Of Sines And Cosines

Which of the following correctly demonstrates the compound angle formula?

Possible Answers:

Correct answer:

Explanation:

The compound angle formula for cosines states that .

Example Question #7 : Complete A Proof Using Sums, Differences, Or Products Of Sines And Cosines

Simplify by applying the compound angle formula:

Possible Answers:

Correct answer:

Explanation:

Using the compound angle formula, we can rewrite each half of the non-coefficient terms in the given expression. Given that   and , substitution yields the following:

 

 

This is the formula for the product of sine and cosine, .

 

Example Question #8 : Complete A Proof Using Sums, Differences, Or Products Of Sines And Cosines

Simplify by applying the compound angle formula:

Possible Answers:

Correct answer:

Explanation:

Using the compound angle formula, we can rewrite each half of the non-coefficient terms in the given expression. Given that  and , substitution yields the following:

 

 

This is the formula for the product of two cosines, .

 

Example Question #9 : Complete A Proof Using Sums, Differences, Or Products Of Sines And Cosines

Using  and the formula for the sum of two sines, rewrite the sum of cosine and sine:

Possible Answers:

Correct answer:

Explanation:

Substitute  for :

 

 

Apply the formula for the sum of two sines, :

 

 

 

Example Question #10 : Complete A Proof Using Sums, Differences, Or Products Of Sines And Cosines

Using  and the formula for the difference of two sines, rewrite the difference of cosine and sine:

Possible Answers:

Correct answer:

Explanation:

Substitute  for :

 

 

Apply the formula for the difference of two sines, .

 

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