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Given sin α = 5 8 , with θ in quadrant I, find cos ( 2 α ) .

cos ( 2 α ) = 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 ) .

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
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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.

( 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 )
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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 ) = 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.

tan ( 2 θ ) = 2 tan θ 1 tan 2 θ Double-angle formula             = 2 tan θ ( 1 tan θ ) ( 1 tan 2 θ ) ( 1 tan θ ) Multiply by a term that results in desired numerator .             = 2 1 tan θ tan 2 θ tan θ             = 2 cot θ tan θ Use reciprocal identity for  1 tan θ .
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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 :

cos ( 2 θ ) = 1 2 sin 2 θ   2 sin 2 θ = 1 cos ( 2 θ )     sin 2 θ = 1 cos ( 2 θ ) 2

Next, we use the formula cos ( 2 θ ) = 2 cos 2 θ 1. Solve for cos 2 θ :

        cos ( 2 θ ) = 2 cos 2 θ 1   1 + cos ( 2 θ ) = 2 cos 2 θ 1 + cos ( 2 θ ) 2 = cos 2 θ

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

tan 2 θ = sin 2 θ cos 2 θ           = 1 cos ( 2 θ ) 2 1 + cos ( 2 θ ) 2 Substitute the reduction formulas.           = ( 1 cos ( 2 θ ) 2 ) ( 2 1 + cos ( 2 θ ) )           = 1 cos ( 2 θ ) 1 + cos ( 2 θ )

Reduction formulas

The reduction formulas    are summarized as follows:

sin 2 θ = 1 cos ( 2 θ ) 2
cos 2 θ = 1 + cos ( 2 θ ) 2
tan 2 θ = 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.

cos 4 x = ( cos 2 x ) 2            = ( 1 + cos ( 2 x ) 2 ) 2 Substitute reduction formula for cos 2 x .            = 1 4 ( 1 + 2 cos ( 2 x ) + cos 2 ( 2 x ) )            = 1 4 + 1 2 cos ( 2 x ) + 1 4 ( 1 + cos 2 ( 2 x ) 2 )  Substitute reduction formula for cos 2 x .            = 1 4 + 1 2 cos ( 2 x ) + 1 8 + 1 8 cos ( 4 x )            = 3 8 + 1 2 cos ( 2 x ) + 1 8 cos ( 4 x )
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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?
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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
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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
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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.
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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?
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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
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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?
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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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