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cos ( 3 x ) cos 3 x = cos x sin 2 x sin x sin ( 2 x )

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For the following exercise, simplify the expression.

tan ( 1 2 x ) + tan ( 1 8 x ) 1 tan ( 1 8 x ) tan ( 1 2 x )

tan ( 5 8 x )

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For the following exercises, find the exact value.

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

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tan ( sin 1 ( 0 ) + sin 1 ( 1 2 ) )

3 3

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Double-Angle, Half-Angle, and Reduction Formulas

For the following exercises, find the exact value.

Find sin ( 2 θ ) , cos ( 2 θ ) , and tan ( 2 θ ) given cos θ = 1 3 and θ is in the interval [ π 2 , π ] .

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Find sin ( 2 θ ) , cos ( 2 θ ) , and tan ( 2 θ ) given sec θ = 5 3 and θ is in the interval [ π 2 , π ] .

24 25 , 7 25 , 24 7

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sec ( 3 π 8 )

2 ( 2 + 2 )

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For the following exercises, use [link] to find the desired quantities.

Image of a right triangle. The base is 24, the height is unknown, and the hypotenuse is 25. The angle opposite the base is labeled alpha, and the remaining acute angle is labeled beta.

sin ( 2 β ) , cos ( 2 β ) , tan ( 2 β ) , sin ( 2 α ) , cos ( 2 α ) ,  and  tan ( 2 α )

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sin ( β 2 ) , cos ( β 2 ) , tan ( β 2 ) , sin ( α 2 ) , cos ( α 2 ) ,  and  tan ( α 2 )

2 10 , 7 2 10 , 1 7 , 3 5 , 4 5 , 3 4

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For the following exercises, prove the identity.

2 cos ( 2 x ) sin ( 2 x ) = cot x tan x

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

cot x cos ( 2 x ) = cot x ( 1 2 sin 2 x )                      = cot x cos x sin x ( 2 ) sin 2 x                      = 2 sin x cos x + cot x                      = sin ( 2 x ) + cot x

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For the following exercises, rewrite the expression with no powers.

cos 2 x sin 4 ( 2 x )

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tan 2 x sin 3 x

10 sin x 5 sin ( 3 x ) + sin ( 5 x ) 8 ( cos ( 2 x ) + 1 )

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Sum-to-Product and Product-to-Sum Formulas

For the following exercises, evaluate the product for the given expression using a sum or difference of two functions. Write the exact answer.

cos ( π 3 ) sin ( π 4 )

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2 sin ( 2 π 3 ) sin ( 5 π 6 )

3 2

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2 cos ( π 5 ) cos ( π 3 )

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For the following exercises, evaluate the sum by using a product formula. Write the exact answer.

sin ( π 12 ) sin ( 7 π 12 )

2 2

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cos ( 5 π 12 ) + cos ( 7 π 12 )

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For the following exercises, change the functions from a product to a sum or a sum to a product.

sin ( 9 x ) cos ( 3 x )

1 2 ( sin ( 6 x ) + sin ( 12 x ) )

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cos ( 7 x ) cos ( 12 x )

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sin ( 11 x ) + sin ( 2 x )

2 sin ( 13 2 x ) cos ( 9 2 x )

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cos ( 6 x ) + cos ( 5 x )

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Solving Trigonometric Equations

For the following exercises, find all exact solutions on the interval [ 0 , 2 π ) .

tan x + 1 = 0

3 π 4 , 7 π 4

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2 sin ( 2 x ) + 2 = 0

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For the following exercises, find all exact solutions on the interval [ 0 , 2 π ) .

2 sin 2 x sin x = 0

0 , π 6 , 5 π 6 , π

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cos 2 x cos x 1 = 0

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2 sin 2 x + 5 sin x + 3 = 0

3 π 2

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cos x 5 sin ( 2 x ) = 0

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1 sec 2 x + 2 + sin 2 x + 4 cos 2 x = 0

No solution

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For the following exercises, simplify the equation algebraically as much as possible. Then use a calculator to find the solutions on the interval [ 0 , 2 π ) . Round to four decimal places.

3 cot 2 x + cot x = 1

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csc 2 x 3 csc x 4 = 0

0.2527 , 2.8889 , 4.7124

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For the following exercises, graph each side of the equation to find the zeroes on the interval [ 0 , 2 π ) .

20 cos 2 x + 21 cos x + 1 = 0

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sec 2 x 2 sec x = 15

1.3694 ,   1.9106 ,   4.3726 ,   4.9137

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Modeling with Trigonometric Equations

For the following exercises, graph the points and find a possible formula for the trigonometric values in the given table.

x 0 1 2 3 4 5
y 1 6 11 6 1 6
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x y
0 2
1 1
2 2
3 5
4 2
5 1

3 sin ( x π 2 ) 2

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x y
3 3 + 2 2
2 3
1 2 2 1
0 1
1 3 2 2
2 1
3 −1 2 2
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A man with his eye level 6 feet above the ground is standing 3 feet away from the base of a 15-foot vertical ladder. If he looks to the top of the ladder, at what angle above horizontal is he looking?

71.6

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Using the ladder from the previous exercise, if a 6-foot-tall construction worker standing at the top of the ladder looks down at the feet of the man standing at the bottom, what angle from the horizontal is he looking?

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Questions & Answers

if three forces F1.f2 .f3 act at a point on a Cartesian plane in the daigram .....so if the question says write down the x and y components ..... I really don't understand
Syamthanda Reply
hey , can you please explain oxidation reaction & redox ?
Boitumelo Reply
hey , can you please explain oxidation reaction and redox ?
Boitumelo
for grade 12 or grade 11?
Sibulele
the value of V1 and V2
Tumelo Reply
advantages of electrons in a circuit
Rethabile Reply
we're do you find electromagnetism past papers
Ntombifuthi
what a normal force
Tholulwazi Reply
it is the force or component of the force that the surface exert on an object incontact with it and which acts perpendicular to the surface
Sihle
what is physics?
Petrus Reply
what is the half reaction of Potassium and chlorine
Anna Reply
how to calculate coefficient of static friction
Lisa Reply
how to calculate static friction
Lisa
How to calculate a current
Tumelo
how to calculate the magnitude of horizontal component of the applied force
Mogano
How to calculate force
Monambi
a structure of a thermocouple used to measure inner temperature
Anna Reply
a fixed gas of a mass is held at standard pressure temperature of 15 degrees Celsius .Calculate the temperature of the gas in Celsius if the pressure is changed to 2×10 to the power 4
Amahle Reply
How is energy being used in bonding?
Raymond Reply
what is acceleration
Syamthanda Reply
a rate of change in velocity of an object whith respect to time
Khuthadzo
how can we find the moment of torque of a circular object
Kidist
Acceleration is a rate of change in velocity.
Justice
t =r×f
Khuthadzo
how to calculate tension by substitution
Precious Reply
hi
Shongi
hi
Leago
use fnet method. how many obects are being calculated ?
Khuthadzo
khuthadzo hii
Hulisani
how to calculate acceleration and tension force
Lungile Reply
you use Fnet equals ma , newtoms second law formula
Masego
please help me with vectors in two dimensions
Mulaudzi Reply
how to calculate normal force
Mulaudzi
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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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