9.4 Sum-to-product and product-to-sum formulas (Page 3/6)

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Verifying the identity using double-angle formulas and reciprocal identities

Verify the identity $\csc^{2} θ - 2 = \frac{\cos (2 θ)}{\sin^{2} θ} .$

For verifying this equation, we are bringing together several of the identities. We will use the double-angle formula and the reciprocal identities. We will work with the right side of the equation and rewrite it until it matches the left side.

\begin{matrix} \frac{\cos (2 θ)}{\sin^{2} θ} & = & \frac{1 - 2 \sin^{2} θ}{\sin^{2} θ} \\ = & \frac{1}{\sin^{2} θ} - \frac{2 \sin^{2} θ}{\sin^{2} θ} \\ = & \csc^{2} θ - 2 \end{matrix}

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Verify the identity $\tan θ \cot θ - \cos^{2} θ = \sin^{2} θ .$

$\begin{matrix} \tan θ \cot θ - \cos^{2} θ & = & (\frac{\sin θ}{\cos θ}) (\frac{\cos θ}{\sin θ}) - \cos^{2} θ \\ = & 1 - \cos^{2} θ \\ = & \sin^{2} θ \end{matrix}$

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Media

Access these online resources for additional instruction and practice with the product-to-sum and sum-to-product identities.

Key equations

Product-to-sum Formulas	$\begin{matrix} \cos α \cos β & = & \frac{1}{2} [\cos (α - β) + \cos (α + β)] \\ \sin α \cos β & = & \frac{1}{2} [\sin (α + β) + \sin (α - β)] \\ \sin α \sin β & = & \frac{1}{2} [\cos (α - β) - \cos (α + β)] \\ \cos α \sin β & = & \frac{1}{2} [\sin (α + β) - \sin (α - β)] \end{matrix}$
Sum-to-product Formulas	$\begin{matrix} \sin α + \sin β & = & 2 \sin (\frac{α + β}{2}) \cos (\frac{α - β}{2}) \\ \sin α - \sin β & = & 2 \sin (\frac{α - β}{2}) \cos (\frac{α + β}{2}) \\ \cos α - \cos β & = & - 2 \sin (\frac{α + β}{2}) \sin (\frac{α - β}{2}) \\ \cos α + \cos β & = & 2 \cos (\frac{α + β}{2}) \cos (\frac{α - β}{2}) \end{matrix}$

Key concepts

From the sum and difference identities, we can derive the product-to-sum formulas and the sum-to-product formulas for sine and cosine.
We can use the product-to-sum formulas to rewrite products of sines, products of cosines, and products of sine and cosine as sums or differences of sines and cosines. See [link] , [link] , and [link] .
We can also derive the sum-to-product identities from the product-to-sum identities using substitution.
We can use the sum-to-product formulas to rewrite sum or difference of sines, cosines, or products sine and cosine as products of sines and cosines. See [link] .
Trigonometric expressions are often simpler to evaluate using the formulas. See [link] .
The identities can be verified using other formulas or by converting the expressions to sines and cosines. To verify an identity, we choose the more complicated side of the equals sign and rewrite it until it is transformed into the other side. See [link] and [link] .

Section exercises

Verbal

Starting with the product to sum formula $\sin α \cos β = \frac{1}{2} [\sin (α + β) + \sin (α - β)],$ explain how to determine the formula for $\cos α \sin β .$

Substitute $α$ into cosine and $β$ into sine and evaluate.

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Provide two different methods of calculating $\cos (195°) \cos (105°),$ one of which uses the product to sum. Which method is easier?

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Describe a situation where we would convert an equation from a sum to a product and give an example.

Answers will vary. There are some equations that involve a sum of two trig expressions where when converted to a product are easier to solve. For example: $\frac{\sin (3 x) + \sin x}{\cos x} = 1.$ When converting the numerator to a product the equation becomes: $\frac{2 \sin (2 x) \cos x}{\cos x} = 1$

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Describe a situation where we would convert an equation from a product to a sum, and give an example.

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Algebraic

For the following exercises, rewrite the product as a sum or difference.

$16 \sin (16 x) \sin (11 x)$

$8 (\cos (5 x) - \cos (27 x))$

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$20 \cos (36 t) \cos (6 t)$

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$2 \sin (5 x) \cos (3 x)$

$\sin (2 x) + \sin (8 x)$

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$10 \cos (5 x) \sin (10 x)$

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$\sin (- x) \sin (5 x)$

$\frac{1}{2} (\cos (6 x) - \cos (4 x))$

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$\sin (3 x) \cos (5 x)$

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For the following exercises, rewrite the sum or difference as a product.

Questions & Answers

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Step 1: Find the mean. To find the mean, add up all the scores, then divide them by the number of scores. ... Step 2: Find each score's deviation from the mean. ... Step 3: Square each deviation from the mean. ... Step 4: Find the sum of squares. ... Step 5: Divide the sum of squares by n – 1 or N.

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Bhartdarshan' is an internet-based travel agency wherein customer can see videos of the cities they plant to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400 a. what is the probability of getting more than 12,000 hits? b. what is the probability of getting fewer than 9,000 hits?

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Bhartdarshan'is an internet-based travel agency wherein customer can see videos of the cities they plan to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400. a. What is the probability of getting more than 12,000 hits

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Source: OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6

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