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

<< Chapter < Page

Precalculus

Page 2 / 8

Chapter >> Page >

Try It

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

$\cos (2 α) = \frac{7}{32}$

Got questions? Get instant answers now!

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}

Got questions? Get instant answers now!

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}

Got questions? Get instant answers now!

Try It

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

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

Got questions? Get instant answers now!

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}

Got questions? Get instant answers now!

Try It

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

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

Got questions? Get instant answers now!

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}

Got questions? Get instant answers now!

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?

Aislinn Reply

tijani

what is titration

John Reply

what is physics

Siyaka Reply

A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance

Jude Reply

Can you compute that for me. Ty

Jude

what is the dimension formula of energy?

David Reply

what is viscosity?

David

what is inorganic

emma Reply

what is chemistry

Youesf Reply

what is inorganic

emma

Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes

Adjei

please, I'm a physics student and I need help in physics

Adjanou

chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.

Pedro

A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity

Krampah Reply

2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.

Sahid Reply

you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s

Samuel Reply

can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?

Joseph Reply

Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time

Joseph

Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing

Joseph

"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true

Ryan

what's motion

Maurice Reply

what are the types of wave

Maurice

answer

Magreth

progressive wave

Magreth

hello friend how are you

Muhammad Reply

fine, how about you?

Mohammed

Mujahid

A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?

yasuo Reply

Who can show me the full solution in this problem?

Reofrir Reply

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Practice Key Terms 3

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

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

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask

	4 Using half-angle formulas to find exact values By OpenStax Read Online Course
	2 Using double-angle formulas to verify identities By OpenStax Read Online Course
	Anthropology Marriage Family Household By Richley Crapo Start Assignment
	10 AP 10 Muscle Tissue MCQ Quiz By OpenStax Start Quiz
	Principles of macroeconomics for ap® courses By OpenStax Read Online Course
	34 Biology 34 Animal Nutrition Digestive System By OpenStax Start Quiz
	1 AP 01 Human Body Anatomy Physiology Essay By OpenStax Start Flashcards
	Introduction to Mechanics MCQ By Saylor Foundation Start Quiz
	1 BOD - DERMATOLOGY - By Brooke Delaney Start Exam
	Macroeconomics MCQ By Candice Butts Start Quiz
	7 AP 07 Axial Skeleton Essay By OpenStax Start Flashcards
	39 Biology 39 The Respiratory System MCQ By OpenStax Start Quiz