7.3 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{array}{l} \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{array}

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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 establish an identity

Establish 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{array}{l} {(\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{array}

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Try It

Establish 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{array}{l} \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})} \begin{matrix}  \end{matrix} & 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{array}

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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{array}{l} \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{array}

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

\begin{array}{l} \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{array}

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

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

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{array}{l} \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}) \begin{matrix}  \end{matrix} & {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{array}

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Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.

Kate Reply

To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.

Muhammed

What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?

Kate Reply

what is the change in momentum of a body?

Eunice Reply

what is a capacitor?

Raymond Reply

Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.

Gautam

A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate

Maria Reply

please solve

Sharon

8m/s²

Aishat

What is Thermodynamics

Muordit

velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.

Mehmet

A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat

Saheed Reply

50 m/s due south east

Someone

which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer

Ramon Reply

I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.

Someone

Scratch that

Someone

temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....

Someone

about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle

Someone

definitely of physics

Haryormhidey Reply

how many start and codon

Esrael Reply

what is field

Felix Reply

physics, biology and chemistry this is my Field

ALIYU

field is a region of space under the influence of some physical properties

Collete

what is ogarnic chemistry

WISDOM Reply

determine the slope giving that 3y+ 2x-14=0

WISDOM

Another formula for Acceleration

Belty Reply

a=v/t. a=f/m a

IHUMA

innocent

Adah

pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?

Nassze Reply

how do lnternal energy measures

Esrael

Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.

JALLAH Reply

No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.

Dlovan

Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?

Robert

like charges repel while unlike charges atttact

Raymond

What is specific heat capacity

Destiny Reply

Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).

AI-Robot

specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin

ROKEEB

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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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