9.3 Double-angle, half-angle, and reduction formulas (Page 2/8)

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Given $\sin α = \frac{5}{8},$ with $θ$ in quadrant I, find $\cos (2 α) .$

$\cos (2 α) = \frac{7}{32}$

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Using the double-angle formula for cosine without exact values

Use the double-angle formula for cosine to write $\cos (6 x)$ in terms of $\cos (3 x) .$

\begin{matrix} \cos (6 x) & = & \cos (3 x + 3 x) \\ = & \cos 3 x \cos 3 x - \sin 3 x \sin 3 x \\ = & \cos^{2} 3 x - \sin^{2} 3 x \end{matrix}

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Using double-angle formulas to verify identities

Establishing identities using the double-angle formulas is performed using the same steps we used to derive the sum and difference formulas. Choose the more complicated side of the equation and rewrite it until it matches the other side.

Using the double-angle formulas to verify an identity

Verify the following identity using double-angle formulas:

1 + \sin (2 θ) = {(\sin θ + \cos θ)}^{2}

We will work on the right side of the equal sign and rewrite the expression until it matches the left side.

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

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Verify the identity: $\cos^{4} θ - \sin^{4} θ = \cos (2 θ) .$

$\cos^{4} θ - \sin^{4} θ = (\cos^{2} θ + \sin^{2} θ) (\cos^{2} θ - \sin^{2} θ) = \cos (2 θ)$

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Verifying a double-angle identity for tangent

Verify the identity:

\tan (2 θ) = \frac{2}{\cot θ - \tan θ}

In this case, we will work with the left side of the equation and simplify or rewrite until it equals the right side of the equation.

\begin{matrix} \tan (2 θ) & = & \frac{2 \tan θ}{1 - \tan^{2} θ} & Double-angle formula \\ = & \frac{2 \tan θ (\frac{1}{\tan θ})}{(1 - \tan^{2} θ) (\frac{1}{\tan θ})} & Multiply by a term that results in desired numerator . \\ = & \frac{2}{\frac{1}{\tan θ} - \frac{\tan^{2} θ}{\tan θ}} \\ = & \frac{2}{\cot θ - \tan θ} & Use reciprocal identity for \frac{1}{\tan θ} . \end{matrix}

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Try It

Verify the identity: $\cos (2 θ) \cos θ = \cos^{3} θ - \cos θ \sin^{2} θ .$

$\cos (2 θ) \cos θ = (\cos^{2} θ - \sin^{2} θ) \cos θ = \cos^{3} θ - \cos θ \sin^{2} θ$

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Use reduction formulas to simplify an expression

The double-angle formulas can be used to derive the reduction formulas , which are formulas we can use to reduce the power of a given expression involving even powers of sine or cosine. They allow us to rewrite the even powers of sine or cosine in terms of the first power of cosine. These formulas are especially important in higher-level math courses, calculus in particular. Also called the power-reducing formulas, three identities are included and are easily derived from the double-angle formulas.

We can use two of the three double-angle formulas for cosine to derive the reduction formulas for sine and cosine. Let’s begin with $\cos (2 θ) = 1 - 2 \sin^{2} θ .$ Solve for $\sin^{2} θ :$

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

Next, we use the formula $\cos (2 θ) = 2 \cos^{2} θ - 1.$ Solve for $\cos^{2} θ :$

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

The last reduction formula is derived by writing tangent in terms of sine and cosine:

\begin{matrix} \tan^{2} θ & = & \frac{\sin^{2} θ}{\cos^{2} θ} \\ = & \frac{\frac{1 - \cos (2 θ)}{2}}{\frac{1 + \cos (2 θ)}{2}} & Substitute the reduction formulas . \\ = & (\frac{1 - \cos (2 θ)}{2}) (\frac{2}{1 + \cos (2 θ)}) \\ = & \frac{1 - \cos (2 θ)}{1 + \cos (2 θ)} \end{matrix}

A General Note

Reduction formulas

The reduction formulas are summarized as follows:

\sin^{2} θ = \frac{1 - \cos (2 θ)}{2}

\cos^{2} θ = \frac{1 + \cos (2 θ)}{2}

\tan^{2} θ = \frac{1 - \cos (2 θ)}{1 + \cos (2 θ)}

Writing an equivalent expression not containing powers greater than 1

Write an equivalent expression for $\cos^{4} x$ that does not involve any powers of sine or cosine greater than 1.

We will apply the reduction formula for cosine twice.

\begin{matrix} \cos^{4} x & = & {(\cos^{2} x)}^{2} \\ = & {(\frac{1 + \cos (2 x)}{2})}^{2} & {Substitute reduction formula for cos}^{2} x . \\ = & \frac{1}{4} (1 + 2 \cos (2 x) + \cos^{2} (2 x)) \\ = & \frac{1}{4} + \frac{1}{2} \cos (2 x) + \frac{1}{4} (\frac{1 + \cos 2 (2 x)}{2}) & {Substitute reduction formula for cos}^{2} x . \\ = & \frac{1}{4} + \frac{1}{2} \cos (2 x) + \frac{1}{8} + \frac{1}{8} \cos (4 x) \\ = & \frac{3}{8} + \frac{1}{2} \cos (2 x) + \frac{1}{8} \cos (4 x) \end{matrix}

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

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