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Using the power-reducing formulas to prove an identity

Use the power-reducing formulas to prove

sin 3 ( 2 x ) = [ 1 2 sin ( 2 x ) ] [ 1 cos ( 4 x ) ]

We will work on simplifying the left side of the equation:

sin 3 ( 2 x ) = [ sin ( 2 x ) ] [ sin 2 ( 2 x ) ]               = sin ( 2 x ) [ 1 cos ( 4 x ) 2 ] Substitute the power-reduction formula .               = sin ( 2 x ) ( 1 2 ) [ 1 cos ( 4 x ) ]               = 1 2 [ sin ( 2 x ) ] [ 1 cos ( 4 x ) ]
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Use the power-reducing formulas to prove that 10 cos 4 x = 15 4 + 5 cos ( 2 x ) + 5 4 cos ( 4 x ) .

10 cos 4 x = 10 cos 4 x = 10 ( cos 2 x ) 2              = 10 [ 1 + cos ( 2 x ) 2 ] 2 Substitute reduction formula for cos 2 x .              = 10 4 [ 1 + 2 cos ( 2 x ) + cos 2 ( 2 x ) ]              = 10 4 + 10 2 cos ( 2 x ) + 10 4 ( 1 + cos 2 ( 2 x ) 2 ) Substitute reduction formula for cos 2 x .              = 10 4 + 10 2 cos ( 2 x ) + 10 8 + 10 8 cos ( 4 x )              = 30 8 + 5 cos ( 2 x ) + 10 8 cos ( 4 x )              = 15 4 + 5 cos ( 2 x ) + 5 4 cos ( 4 x )

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Using half-angle formulas to find exact values

The next set of identities is the set of half-angle formulas    , which can be derived from the reduction formulas and we can use when we have an angle that is half the size of a special angle. If we replace θ with α 2 , the half-angle formula for sine is found by simplifying the equation and solving for sin ( α 2 ) . Note that the half-angle formulas are preceded by a ± sign. This does not mean that both the positive and negative expressions are valid. Rather, it depends on the quadrant in which α 2 terminates.

The half-angle formula for sine is derived as follows:

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

To derive the half-angle formula for cosine, we have

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

For the tangent identity, we have

    tan 2 θ = 1 cos ( 2 θ ) 1 + cos ( 2 θ ) tan 2 ( α 2 ) = 1 cos ( 2 α 2 ) 1 + cos ( 2 α 2 )              = 1 cos α 1 + cos α    tan ( α 2 ) = ± 1 cos α 1 + cos α

Half-angle formulas

The half-angle formulas    are as follows:

sin ( α 2 ) = ± 1 cos α 2
cos ( α 2 ) = ± 1 + cos α 2
tan ( α 2 ) = ± 1 cos α 1 + cos α = sin α 1 + cos α = 1 cos α sin α

Using a half-angle formula to find the exact value of a sine function

Find sin ( 15 ) using a half-angle formula.

Since 15 = 30 2 , we use the half-angle formula for sine:

sin 30 2 = 1 cos 30 2             = 1 3 2 2             = 2 3 2 2             = 2 3 4             = 2 3 2
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Given the tangent of an angle and the quadrant in which the angle lies, find the exact values of trigonometric functions of half of the angle.

  1. Draw a triangle to represent the given information.
  2. Determine the correct half-angle formula.
  3. Substitute values into the formula based on the triangle.
  4. Simplify.

Finding exact values using half-angle identities

Given that tan α = 8 15 and α lies in quadrant III, find the exact value of the following:

  1. sin ( α 2 )
  2. cos ( α 2 )
  3. tan ( α 2 )

Using the given information, we can draw the triangle shown in [link] . Using the Pythagorean Theorem, we find the hypotenuse to be 17. Therefore, we can calculate sin α = 8 17 and cos α = 15 17 .

Diagram of a triangle in the x,y-plane. The vertices are at the origin, (-15,0), and (-15,-8). The angle at the origin is alpha. The angle formed by the side (-15,-8) to (-15,0) forms a right angle with the x axis. The hypotenuse across from the right angle is length 17.
  1. Before we start, we must remember that, if α is in quadrant III, then 180° < α < 270° , so 180° 2 < α 2 < 270° 2 . This means that the terminal side of α 2 is in quadrant II, since 90° < α 2 < 135° .

    To find sin α 2 , we begin by writing the half-angle formula for sine. Then we substitute the value of the cosine we found from the triangle in [link] and simplify.

    sin α 2 = ± 1 cos α 2          = ± 1 ( 15 17 ) 2          = ± 32 17 2          = ± 32 17 1 2          = ± 16 17          = ± 4 17          = 4 17 17

    We choose the positive value of sin α 2 because the angle terminates in quadrant II and sine is positive in quadrant II.

  2. To find cos α 2 , we will write the half-angle formula for cosine, substitute the value of the cosine we found from the triangle in [link] , and simplify.
    cos α 2 = ± 1 + cos α 2          = ± 1 + ( 15 17 ) 2          = ± 2 17 2          = ± 2 17 1 2          = ± 1 17          = 17 17

    We choose the negative value of cos α 2 because the angle is in quadrant II because cosine is negative in quadrant II.

  3. To find tan α 2 , we write the half-angle formula for tangent. Again, we substitute the value of the cosine we found from the triangle in [link] and simplify.
    tan α 2 = ± 1 cos α 1 + cos α          = ± 1 ( 15 17 ) 1 + ( 15 17 )          = ± 32 17 2 17          = ± 32 2          = 16          = 4

    We choose the negative value of tan α 2 because α 2 lies in quadrant II, and tangent is negative in quadrant II.

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Questions & Answers

The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
Rima Reply
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
Jhon Reply
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
austin Reply
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
more than 6000
Robert
can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
how to understand calculus?
Jenica Reply
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
Practice Key Terms 3

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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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