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Section exercises

Verbal

Explain the basis for the cofunction identities and when they apply.

The cofunction identities apply to complementary angles. Viewing the two acute angles of a right triangle, if one of those angles measures x , the second angle measures π 2 x . Then sin x = cos ( π 2 x ) . The same holds for the other cofunction identities. The key is that the angles are complementary.

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Is there only one way to evaluate cos ( 5 π 4 ) ? Explain how to set up the solution in two different ways, and then compute to make sure they give the same answer.

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Explain to someone who has forgotten the even-odd properties of sinusoidal functions how the addition and subtraction formulas can determine this characteristic for f ( x ) = sin ( x ) and g ( x ) = cos ( x ) . (Hint: 0 x = x )

sin ( x ) = sin x , so sin x is odd. cos ( x ) = cos ( 0 x ) = cos x , so cos x is even.

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Algebraic

For the following exercises, find the exact value.

sin ( 11 π 12 )

6 2 4

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tan ( 19 π 12 )

2 3

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For the following exercises, rewrite in terms of sin x and cos x .

sin ( x 3 π 4 )

2 2 sin x 2 2 cos x

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cos ( x + 2 π 3 )

1 2 cos x 3 2 sin x

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For the following exercises, simplify the given expression.

sec ( π 2 θ )

csc θ

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tan ( π 2 x )

cot x

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sin ( 2 x ) cos ( 5 x ) sin ( 5 x ) cos ( 2 x )

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tan ( 3 2 x ) tan ( 7 5 x ) 1 + tan ( 3 2 x ) tan ( 7 5 x )

tan ( x 10 )

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For the following exercises, find the requested information.

Given that sin a = 2 3 and cos b = 1 4 , with a and b both in the interval [ π 2 , π ) , find sin ( a + b ) and cos ( a b ) .

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Given that sin a = 4 5 , and cos b = 1 3 , with a and b both in the interval [ 0 , π 2 ) , find sin ( a b ) and cos ( a + b ) .

sin ( a b ) = ( 4 5 ) ( 1 3 ) ( 3 5 ) ( 2 2 3 ) = 4 6 2 15
cos ( a + b ) = ( 3 5 ) ( 1 3 ) ( 4 5 ) ( 2 2 3 ) = 3 8 2 15

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For the following exercises, find the exact value of each expression.

sin ( cos 1 ( 0 ) cos 1 ( 1 2 ) )

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cos ( cos 1 ( 2 2 ) + sin 1 ( 3 2 ) )

2 6 4

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tan ( sin 1 ( 1 2 ) cos 1 ( 1 2 ) )

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Graphical

For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical.

cos ( π 2 x )

sin x

Graph of y=sin(x) from -2pi to 2pi.
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tan ( π 3 + x )

cot ( π 6 x )

Graph of y=cot(pi/6 - x) from -2pi to pi - in comparison to the usual y=cot(x) graph, this one is reflected across the x-axis and shifted by pi/6.
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tan ( π 4 x )

cot ( π 4 + x )

Graph of y=cot(pi/4 + x) - in comparison to the usual y=cot(x) graph, this one is shifted by pi/4.
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sin ( π 4 + x )

sin x 2 + cos x 2

Graph of y = sin(x) / rad2 + cos(x) / rad2 - it looks like the sin curve shifted by pi/4.
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For the following exercises, use a graph to determine whether the functions are the same or different. If they are the same, show why. If they are different, replace the second function with one that is identical to the first. (Hint: think 2 x = x + x . )

f ( x ) = sin ( 4 x ) sin ( 3 x ) cos x , g ( x ) = sin x cos ( 3 x )

They are the same.

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f ( x ) = cos ( 4 x ) + sin x sin ( 3 x ) , g ( x ) = cos x cos ( 3 x )

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f ( x ) = sin ( 3 x ) cos ( 6 x ) , g ( x ) = sin ( 3 x ) cos ( 6 x )

They are the different, try g ( x ) = sin ( 9 x ) cos ( 3 x ) sin ( 6 x ) .

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f ( x ) = sin ( 4 x ) , g ( x ) = sin ( 5 x ) cos x cos ( 5 x ) sin x

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f ( x ) = sin ( 2 x ) , g ( x ) = 2 sin x cos x

They are the same.

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f ( θ ) = cos ( 2 θ ) , g ( θ ) = cos 2 θ sin 2 θ

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f ( θ ) = tan ( 2 θ ) , g ( θ ) = tan θ 1 + tan 2 θ

They are the different, try g ( θ ) = 2 tan θ 1 tan 2 θ .

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f ( x ) = sin ( 3 x ) sin x , g ( x ) = sin 2 ( 2 x ) cos 2 x cos 2 ( 2 x ) sin 2 x

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f ( x ) = tan ( x ) , g ( x ) = tan x tan ( 2 x ) 1 tan x tan ( 2 x )

They are different, try g ( x ) = tan x tan ( 2 x ) 1 + tan x tan ( 2 x ) .

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Technology

For the following exercises, find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point.

sin ( 195 )

3 1 2 2 ,  or  0.2588

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cos ( 345 )

1 + 3 2 2 , or 0.9659

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Extensions

For the following exercises, prove the identities provided.

tan ( x + π 4 ) = tan x + 1 1 tan x

tan ( x + π 4 ) = tan x + tan ( π 4 ) 1 tan x tan ( π 4 ) = tan x + 1 1 tan x ( 1 ) = tan x + 1 1 tan x

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tan ( a + b ) tan ( a b ) = sin a cos a + sin b cos b sin a cos a sin b cos b

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cos ( a + b ) cos a cos b = 1 tan a tan b

cos ( a + b ) cos a cos b = cos a cos b cos a cos b sin a sin b cos a cos b = 1 tan a tan b

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cos ( x + y ) cos ( x y ) = cos 2 x sin 2 y

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cos ( x + h ) cos x h = cos x cos h 1 h sin x sin h h

cos ( x + h ) cos x h = cos x cosh sin x sinh cos x h = cos x ( cosh 1 ) sin x sinh h = cos x cos h 1 h sin x sin h h

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For the following exercises, prove or disprove the statements.

tan ( u + v ) = tan u + tan v 1 tan u tan v

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tan ( u v ) = tan u tan v 1 + tan u tan v

True

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tan ( x + y ) 1 + tan x tan x = tan x + tan y 1 tan 2 x tan 2 y

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If α , β , and γ are angles in the same triangle, then prove or disprove sin ( α + β ) = sin γ .

True. Note that sin ( α + β ) = sin ( π γ ) and expand the right hand side.

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If α , β , and y are angles in the same triangle, then prove or disprove tan α + tan β + tan γ = tan α tan β tan γ

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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?
Aislinn Reply
cm
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
hi
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
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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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