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

Precalculus

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Proving an identity

Prove the identity:

\frac{\cos (4 t) - \cos (2 t)}{\sin (4 t) + \sin (2 t)} = - \tan t

We will start with the left side, the more complicated side of the equation, and rewrite the expression until it matches the right side.

\begin{array}{l} \frac{\cos (4 t) - \cos (2 t)}{\sin (4 t) + \sin (2 t)} = \frac{- 2 \sin (\frac{4 t + 2 t}{2}) \sin (\frac{4 t - 2 t}{2})}{2 \sin (\frac{4 t + 2 t}{2}) \cos (\frac{4 t - 2 t}{2})} \\ = \frac{- 2 \sin (3 t) \sin t}{2 \sin (3 t) \cos t} \\ = \frac{- 2 \sin (3 t) \sin t}{2 \sin (3 t) \cos t} \\ = - \frac{\sin t}{\cos t} \\ = - \tan t \end{array}

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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{array}{l} \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{array}

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Try It

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

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

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

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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Explain 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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Explain 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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Questions & Answers

Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.

Widad Reply

can you send the book attached ?

Ariel

What is economics

Widad Reply

the study of how humans make choices under conditions of scarcity

AI-Robot

U(x,y) = (x×y)1/2 find mu of x for y

Desalegn Reply

U(x,y) = (xÃy)1/2 find mu of x for y

Desalegn

what is ecnomics

Jan Reply

this is the study of how the society manages it's scarce resources

Belonwu

what is macroeconomic

John Reply

macroeconomic is the branch of economics which studies actions, scale, activities and behaviour of the aggregate economy as a whole.

husaini

etc

husaini

difference between firm and industry

husaini Reply

what's the difference between a firm and an industry

Abdul

firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅

Abdulraufu

Suppose the demand function that a firm faces shifted from Qd  120 3P to Qd  90  3P and the supply function has shifted from QS  20  2P to QS 10  2P . a) Find the effect of this change on price and quantity. b) Which of the changes in demand and supply is higher?

Toofiq Reply

explain standard reason why economic is a science

innocent Reply

factors influencing supply

Petrus Reply

what is economic.

Milan Reply

scares means__________________ends resources. unlimited

Jan

economics is a science that studies human behaviour as a relationship b/w ends and scares means which have alternative uses

Jan

calculate the profit maximizing for demand and supply

Zarshad Reply

Why qualify 28 supplies

Milan

what are explicit costs

Nomsa Reply

out-of-pocket costs for a firm, for example, payments for wages and salaries, rent, or materials

AI-Robot

concepts of supply in microeconomics

David Reply

economic overview notes

Amahle Reply

identify a demand and a supply curve

Salome Reply

i don't know

Parul

there's a difference

Aryan

Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods

Israr

Hi Sir please how do u calculate Cross elastic demand and income elastic demand?

Abari

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Source: OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6

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