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Trigonometry : Sum, Difference, and Product Identities

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Example Questions

Example Question #1 : Sum, Difference, And Product Identities

True or false:

$\sin x + \sin y = \sin (x + y)$ $\displaystyle \sin x + \sin y = \sin (x + y)$.

Possible Answers:

Cannot be determined

False

True

Correct answer:

False

Explanation:

The sum of sines is given by the formula $\sin x + \sin y = \2\sin ((x + y)/2) * \cos((x - y)/2)$ $\displaystyle \sin x + \sin y = \2\sin ((x + y)/2) * \cos((x - y)/2)$.

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Example Question #2 : Sum, Difference, And Product Identities

True or false: $\sin x - \sin y = \sin (x - y)$ $\displaystyle \sin x - \sin y = \sin (x - y)$.

Possible Answers:

True

False

Cannot be determined

Correct answer:

False

Explanation:

The difference of sines is given by the formula $\sin x - \sin y = \2\sin ((x - y)/2) * \cos((x + y)/2)$ $\displaystyle \sin x - \sin y = \2\sin ((x - y)/2) * \cos((x + y)/2)$.

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Example Question #3 : Sum, Difference, And Product Identities

True or false: $\cos x + \cos y = \cos (x + y)$ $\displaystyle \cos x + \cos y = \cos (x + y)$.

Possible Answers:

Cannot be determined

True

False

Correct answer:

False

Explanation:

The sum of cosines is given by the formula $\cos x + \cos y = 2\cos ((x + y)/2) * \cos((x - y)/2)$ $\displaystyle \cos x + \cos y = 2\cos ((x + y)/2) * \cos((x - y)/2)$.

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Example Question #4 : Sum, Difference, And Product Identities

True or false: $\cos x - \cos y = \cos (x - y)$ $\displaystyle \cos x - \cos y = \cos (x - y)$.

Possible Answers:

True

False

Cannot be determined

Correct answer:

False

Explanation:

The difference of cosines is given by the formula $\cos x - \cos y = 2\sin ((x + y)/2) * \sin((y - x)/2)$ $\displaystyle \cos x - \cos y = 2\sin ((x + y)/2) * \sin((y - x)/2)$.

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Example Question #5 : Sum, Difference, And Product Identities

Which of the following correctly demonstrates the compound angle formula?

Possible Answers:

$\sin(x - y) = sinxsiny - cosxcosy$ $\displaystyle \sin(x - y) = sinxsiny - cosxcosy$

$\sin(x + y) = sinxcosy - cosxsiny$ $\displaystyle \sin(x + y) = sinxcosy - cosxsiny$

$\sin(x + y) = sinxsiny - cosxcosy$ $\displaystyle \sin(x + y) = sinxsiny - cosxcosy$

$\sin(x - y) = sinxcosy - cosxsiny$ $\displaystyle \sin(x - y) = sinxcosy - cosxsiny$

Correct answer:

$\sin(x - y) = sinxcosy - cosxsiny$ $\displaystyle \sin(x - y) = sinxcosy - cosxsiny$

Explanation:

The compound angle formula for sines states that $\sin(x +- y) = sinxcosy +- cosxsiny$ $\displaystyle \sin(x +- y) = sinxcosy +- cosxsiny$.

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Example Question #6 : Sum, Difference, And Product Identities

Which of the following correctly demonstrates the compound angle formula?

Possible Answers:

$\cos(x + y) = sinxsiny - cosxcosy$ $\displaystyle \cos(x + y) = sinxsiny - cosxcosy$

$\cos(x + y) = sinxcosy - cosxsiny$ $\displaystyle \cos(x + y) = sinxcosy - cosxsiny$

$\cos(x - y) = sinxsiny - cosxcosy$ $\displaystyle \cos(x - y) = sinxsiny - cosxcosy$

$\cos(x - y) = sinxcosy - cosxsiny$ $\displaystyle \cos(x - y) = sinxcosy - cosxsiny$

Correct answer:

$\cos(x + y) = sinxsiny - cosxcosy$ $\displaystyle \cos(x + y) = sinxsiny - cosxcosy$

Explanation:

The compound angle formula for cosines states that $\cos(x +- y) = sinxsiny -+ cosxcosy$ $\displaystyle \cos(x +- y) = sinxsiny -+ cosxcosy$.

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

Simplify by applying the compound angle formula:

$\displaystyle (1/2)(sinxcosy - cosxsiny + sinxcosy + cosxsiny)$

Possible Answers:

$\displaystyle cosxsiny$

$\displaystyle sinxsiny$

$\displaystyle sinxcosy$

$\displaystyle cosxcosy$

Correct answer:

$\displaystyle sinxcosy$

Explanation:

Using the compound angle formula, we can rewrite each half of the non-coefficient terms in the given expression. Given that $\sin (x - y) = sinxcosy - cosxsiny$ $\displaystyle \sin (x - y) = sinxcosy - cosxsiny$ and $\sin (x + y) = sinxcosy + cosxsiny$ $\displaystyle \sin (x + y) = sinxcosy + cosxsiny$, substitution yields the following:

$\displaystyle (1/2)(sinxcosy - cosxsiny + sinxcosy + cosxsiny)$

$\displaystyle (1/2)(sin(x - y) + sin(x + y))$

This is the formula for the product of sine and cosine, $sinxcosy = (1/2)(\sin(x - y) + \sin(x + y))$ $\displaystyle sinxcosy = (1/2)(\sin(x - y) + \sin(x + y))$.

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

Simplify by applying the compound angle formula:

$\displaystyle (1/2)(sinxsiny + cosxcosy + sinxsiny - cosxcosy)$

Possible Answers:

$\displaystyle sinxcosy$

$\displaystyle cosxsiny$

$\displaystyle cosxcosy$

$\displaystyle sinxsiny$

Correct answer:

$\displaystyle cosxcosy$

Explanation:

Using the compound angle formula, we can rewrite each half of the non-coefficient terms in the given expression. Given that $\cos(x - y) = sinxsiny + cosxcosy$ $\displaystyle \cos(x - y) = sinxsiny + cosxcosy$ and $\cos(x + y) = sinxsiny - cosxcosy$ $\displaystyle \cos(x + y) = sinxsiny - cosxcosy$, substitution yields the following:

$\displaystyle (1/2)(sinxsiny + cosxcosy + sinxsiny - cosxcosy)$

$(1/2)(\cos(x - y) + \cos(x + y))$ $\displaystyle (1/2)(\cos(x - y) + \cos(x + y))$

This is the formula for the product of two cosines, $cosxcosy = (1/2)(\cos(x - y) + \cos(x + y))$ $\displaystyle cosxcosy = (1/2)(\cos(x - y) + \cos(x + y))$.

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

Using $\sin(90 - x) = \cos x$ $\displaystyle \sin(90 - x) = \cos x$ and the formula for the sum of two sines, rewrite the sum of cosine and sine:

$\cos x + \sin x$ $\displaystyle \cos x + \sin x$

Possible Answers:

$2\sin(90) * \cos(90 - x)$ $\displaystyle 2\sin(90) * \cos(90 - x)$

$2\sin(45) * \cos(45 - x)$ $\displaystyle 2\sin(45) * \cos(45 - x)$

$\sin(45) * \cos(45 - x)$ $\displaystyle \sin(45) * \cos(45 - x)$

$\sin(90) * \cos(90 - x)$ $\displaystyle \sin(90) * \cos(90 - x)$

Correct answer:

$2\sin(45) * \cos(45 - x)$ $\displaystyle 2\sin(45) * \cos(45 - x)$

Explanation:

Substitute $\sin(90 - x)$ $\displaystyle \sin(90 - x)$ for $\cos x$ $\displaystyle \cos x$:

$\cos x + \sin x$ $\displaystyle \cos x + \sin x$

$\sin(90 - x) + \sin x$ $\displaystyle \sin(90 - x) + \sin x$

Apply the formula for the sum of two sines, $\sin x + \sin y = 2\sin ((x + y)/2) * \cos((x - y)/2)$ $\displaystyle \sin x + \sin y = 2\sin ((x + y)/2) * \cos((x - y)/2)$:

$\sin(90 - x) + \sin x$ $\displaystyle \sin(90 - x) + \sin x$

$2\sin((90 - x + x)/2) * \cos((90 - x - x)/2)$ $\displaystyle 2\sin((90 - x + x)/2) * \cos((90 - x - x)/2)$

$2\sin(45) * \cos(45 - x)$ $\displaystyle 2\sin(45) * \cos(45 - x)$

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Example Question #10 : Sum, Difference, And Product Identities

Using $\sin(90 - x) = \cos x$ $\displaystyle \sin(90 - x) = \cos x$ and the formula for the difference of two sines, rewrite the difference of cosine and sine:

$\cos x - \sin x$ $\displaystyle \cos x - \sin x$

Possible Answers:

$2\sin(45 - x) * \cos(45)$ $\displaystyle 2\sin(45 - x) * \cos(45)$

$\sin(45 - x) * \cos(45)$ $\displaystyle \sin(45 - x) * \cos(45)$

$2\sin(45) * \cos(45 - x)$ $\displaystyle 2\sin(45) * \cos(45 - x)$

$\sin(45) * \cos(45 - x)$ $\displaystyle \sin(45) * \cos(45 - x)$

Correct answer:

$2\sin(45 - x) * \cos(45)$ $\displaystyle 2\sin(45 - x) * \cos(45)$

Explanation:

Substitute $\sin(90 - x)$ $\displaystyle \sin(90 - x)$ for $\cos x$ $\displaystyle \cos x$:

$\cos x - \sin x$ $\displaystyle \cos x - \sin x$

$\sin(90 - x) - \sin x$ $\displaystyle \sin(90 - x) - \sin x$

Apply the formula for the difference of two sines, $\sin x - \sin y = 2\sin((x - y)/2) * \cos((x + y)/2)$ $\displaystyle \sin x - \sin y = 2\sin((x - y)/2) * \cos((x + y)/2)$.

$\sin(90 - x) - \sin x$ $\displaystyle \sin(90 - x) - \sin x$

$2\sin((90 - x - x)/2) * \cos((90 - x + x)/2)$ $\displaystyle 2\sin((90 - x - x)/2) * \cos((90 - x + x)/2)$

$2\sin(45 - x) * \cos(45)$ $\displaystyle 2\sin(45 - x) * \cos(45)$

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Trigonometry : Sum, Difference, and Product Identities

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #1 : Sum, Difference, And Product Identities

Example Question #2 : Sum, Difference, And Product Identities

Example Question #3 : Sum, Difference, And Product Identities

Example Question #4 : Sum, Difference, And Product Identities

Example Question #5 : Sum, Difference, And Product Identities

Example Question #6 : Sum, Difference, And Product Identities

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

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

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

Example Question #10 : Sum, Difference, And Product Identities

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