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

find the equation of the tangent to the curve y=2x³-x²+3x+1 at the points x=1 and x=3

Esther Reply

derivative of logarithms function

Iqra Reply

how to solve this question

sidra

ex 2.1 question no 11

khansa

anyone can help me

khansa

question please

Rasul

ex 2.1 question no. 11

khansa

i cant type here

khansa

Find the derivative of g(x)=−3.

Abdullah Reply

any genius online ? I need help!!

Guzorochi Reply

how can i help you?

Pina

need to learn polynomial

Zakariya

i will teach...

nandu

I'm waiting

Zakariya

plz help me in question

Abish

How can I help you?

Tlou

evaluate the following computation (x³-8/x-2)

Murtala Reply

teach me how to solve the first law of calculus.

Uncle Reply

teach me also how to solve the first law of calculus

Bilson

what is differentiation

Ibrahim Reply

only god knows😂

abdulkadir

f(x) = x-2 g(x) = 3x + 5 fog(x)? f(x)/g(x)

Naufal Reply

fog(x)= f(g(x)) = x-2 = 3x+5-2 = 3x+3 f(x)/g(x)= x-2/3x+5

diron

pweding paturo nsa calculus?

jimmy

how to use fundamental theorem to solve exponential

JULIA Reply

find the bounded area of the parabola y^2=4x and y=16x

Omar Reply

what is absolute value means?

Geo Reply

Chicken nuggets

Hugh

🐔

🐔🦃 nuggets

(mathematics) For a complex number a+bi, the principal square root of the sum of the squares of its real and imaginary parts, √a2+b2 . Denoted by | |. The absolute value |x| of a real number x is √x2 , which is equal to x if x is non-negative, and −x if x is negative.

Ismael

find integration of loge x

Game Reply

find the volume of a solid about the y-axis, x=0, x=1, y=0, y=7+x^3

Godwin Reply

how does this work

Brad Reply

Can calculus give the answers as same as other methods give in basic classes while solving the numericals?

Cosmos Reply

log tan (x/4+x/2)

Rohan

please answer

Rohan

y=(x^2 + 3x).(eipix)

Claudia

is this a answer

Ismael

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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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