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Double angle formulas are used to express sines, cosines, tangents, cotangents of an angle with a value of 2 α, using trigonometric functions of the angle α. This article will introduce all double angle formulas with proofs. Examples of application of formulas will be considered. In the final part, the formulas for triple and quadruple angles will be shown.
Yandex.RTB R-A-339285-1
List of double angle formulas
To convert double angle formulas, remember that angles in trigonometry have the form n α notation, where n is natural number, the value of the expression is written without parentheses. Thus, the notation sin n α is considered to have the same meaning as sin (n α) . When denoting sin n α, we have a similar notation (sin α) n. The use of the recording is applicable to everyone trigonometric functions with powers n.
Below are the double angle formulas:
sin 2 α = 2 · sin α · cos α cos 2 α = cos 2 α - sin 2 α , cos 2 α = 1 - 2 · sin 2 α , cos 2 α = 2 · cos 2 α - 1 t g 2 α = 2 t g α 1 - t g 2 α c t g 2 α - c t g 2 α - 1 2 c t g α
Note that these formulas sin and cos are applicable with any value of the angle α. The double angle tangent formula is valid for any value of α, where t g 2 α makes sense, that is, α ≠ π 4 + π 2 · z, z is any integer. The double angle cotangent exists for any α, where c t g 2 α is defined at α ≠ π 2 z.
The cosine of a double angle has the triple notation of a double angle. All of them are applicable.
Proof of double angle formulas
The proof of the formulas starts from the addition formulas. Let's apply the formulas for the sine of the sum:
sin (α + β) = sin α · cos β + cos α · sin β and the cosine of the sum cos (α + β) = cos α · cos β - sin α · sin β. Let's assume that β = α, then we get that
sin (α + α) = sin α · cos α + cos α · sin α = 2 · sin α · cos α and cos (α + α) = cos α · cos α - sin α · sin α = cos 2 α - sin 2 α
Thus, the formulas for the sine and cosine of the double angle sin 2 α = 2 · sin α · cos α and cos 2 α = cos 2 α - sin 2 α are proven.
Rest cos formulas 2 α = 1 - 2 · sin 2 α and cos 2 α = 2 · cos 2 α - 1 lead to mind cos 2 α = cos 2 α = cos 2 α - sin 2 α, when replacing 1 with the sum of squares according to the main identity sin 2 α + cos 2 α = 1 . We get that sin 2 α + cos 2 α = 1. So 1 - 2 sin 2 α = sin 2 α + cos 2 α - 2 sin 2 α = cos 2 α - sin 2 α and 2 cos 2 α - 1 = 2 cos 2 α - (sin 2 α + cos 2 α) = cos 2 α - sin 2 α.
To prove the formulas for the double angle of tangent and cotangent, we apply the equalities t g 2 α = sin 2 α cos 2 α and c t g 2 α = cos 2 α sin 2 α. After the transformation, we obtain that t g 2 α = sin 2 α cos 2 α = 2 · sin α · cos α cos 2 α - sin 2 α and c t g 2 α = cos 2 α sin 2 α = cos 2 α - sin 2 α 2 · sin α · cos α . Divide the expression by cos 2 α, where cos 2 α ≠ 0 with any value of α when t g α is defined. We divide another expression by sin 2 α, where sin 2 α ≠ 0 with any values of α, when c t g 2 α makes sense. To prove the double angle formula for tangent and cotangent, we substitute and get:
Most Frequently Asked Questions
Is it possible to make a stamp on a document according to the sample provided? Answer Yes, it's possible. Send a scanned copy or a good quality photo to our email address, and we will make the necessary duplicate.
What types of payment do you accept?
Answer You can pay for the document upon receipt by the courier, after checking the correctness of completion and quality of execution of the diploma. This can also be done at the office of postal companies offering cash on delivery services.
All terms of delivery and payment for documents are described in the “Payment and Delivery” section. We are also ready to listen to your suggestions regarding the terms of delivery and payment for the document.
Can I be sure that after placing an order you will not disappear with my money? Answer We have quite a long experience in the field of diploma production. We have several websites that are constantly updated. Our specialists work in different parts of the country, producing over 10 documents a day. Over the years, our documents have helped many people solve employment problems or move to higher-paying jobs. We have earned trust and recognition among clients, so there is absolutely no reason for us to do this. Moreover, this is simply impossible to do physically: you pay for your order the moment you receive it in your hands, there is no prepayment.
Can I order a diploma from any university? Answer In general, yes. We have been working in this field for almost 12 years. During this time, an almost complete database of documents issued by almost all universities in the country and for different years of issue was formed. All you need is to select a university, specialty, document, and fill out the order form.
What to do if you find typos and errors in a document?
Answer When receiving a document from our courier or postal company, we recommend that you carefully check all the details. If a typo, error or inaccuracy is discovered, you have the right not to pick up the diploma, but you must indicate the detected defects personally to the courier or in writing by sending an email.
We will correct the document as soon as possible and resend it to the specified address. Of course, shipping will be paid by our company.
To avoid such misunderstandings, before filling out the original form, we email the customer a mock-up of the future document for checking and approval of the final version. Before sending the document by courier or mail, we also take additional photos and videos (including in ultraviolet light) so that you have a clear idea of what you will receive in the end.
What should I do to order a diploma from your company?
Answer To order a document (certificate, diploma, academic certificate, etc.), you must fill out the online order form on our website or provide your email so that we can send you an application form, which you need to fill out and send back to us.
If you do not know what to indicate in any field of the order form/questionnaire, leave them blank. Therefore, we will clarify all the missing information over the phone.
Latest reviews
Alexei:
I needed to acquire a diploma to get a job as a manager. And the most important thing is that I have both experience and skills, but I can’t get a job without a document. Once I came across your site, I finally decided to buy a diploma. The diploma was completed in 2 days!! Now I have a job that I never dreamed of before!! Thank you!
Very often, in problems C1 from the Unified State Examination in mathematics, students are asked to solve a trigonometric equation containing the double angle formula.
Today we will again analyze problem C1 and, in particular, we will analyze quite non-standard example, which simultaneously contained both the double angle formula and even homogeneous equation. So:
Solve the equation. Find the roots of this equation, belonging to the interval:
sinx+ sin2 x 2 −cos2 x 2 ,x∈ [ −2 π ;− π 2 ]
\sin x+\frac(((\sin )^(2))x)(2)-\frac(((\cos )^(2))x)(2),x\in \left[ -2\ text( )\!\!\pi\!\!\text( );-\frac(\text( )\!\!\pi\!\!\text( ))(2) \right]
Useful formulas for solving
First of all, I would like to remind you that all C1 tasks are solved according to the same scheme. First of all, the original construction must be transformed into an expression that contains a sine, cosine or tangent:
sinx=a
cosx=a
tgx=a
This is precisely the main difficulty of task C1. The fact is that each specific expression requires its own calculations, with the help of which you can move from the source code to such simple constructions. In our case, this is the double angle formula. Let me write it down:
cos2x= cos2 x− sin2 x
\cos 2x=((\cos )^(2))x-((\sin )^(2))x
However, in our task there is no cos2 x((\cos )^(2))x or sin2 x((\sin )^(2))x, but there is sin2 x 2 \frac(((\sin )^(2))x)(2) and cos2 x 2 \frac(((\cos )^(2))x)(2).
Solving the problem
What to do with these calculations? Let's cheat a little and introduce a new variable into our formulas for the sine and cosine of a double angle:
x= t 2
We will write the following construction with sine and cosine:
cos2⋅ t 2=cos2 t 2 −sin2 t 2
\cos 2\cdot \frac(t)(2)=\frac(((\cos )^(2))t)(2)-\frac(((\sin )^(2))t)(2 )
Or in other words:
cost= cos2 t 2 −sin2 t 2
\cos t=\frac(((\cos )^(2))t)(2)-\frac(((\sin )^(2))t)(2)
Let's return to our original task. Let's sin2 x 2 \frac(((\sin )^(2))x)(2) move to the right:
sinx= cos2 x 2 −sin2 x 2
\sin x=\frac(((\cos )^(2))x)(2)-\frac(((\sin )^(2))x)(2)
On the right are exactly the same calculations that we just recorded. Let's convert them:
sinx=cosx
And now attention: before us is a homogeneous trigonometric equation of the first degree. Look, we don't have any terms that are just numbers and just x x, we only have sine and cosine. Also, we do not have quadratic trigonometric functions, all functions go to the first degree. How are such designs solved? First of all, let's assume that cosx=0\cos x=0.
Let's substitute this value into the main trigonometric identity:
sin2 x+ cos2 x=1
((\sin )^(2))x+((\cos )^(2))x=1
sin2 x+0=1
((\sin )^(2))x+0=1
sinx=±1
If we substitute these numbers, 0 and ±1, into the original construction, we get the following:
±1 = 0
\pm 1\text( )=\text( )0
We got complete nonsense. Therefore, our assumption is that cosx=0\cos x=0 is incorrect, cosx\cos x cannot be 0 in this expression. And if cosx\cos x is not equal to 0, then let's divide both sides by cosx\cos x:
sinxcosx=1
\frac(\sin x)(\cos x)=1
sinxcosx=tgx
\frac(\sin x)(\cos x)=tgx
tgx=1
And now we have the long-awaited simplest expression of the form tgx=a tgx=a. Great, let's solve it. This is the table value:
x= π 4 + π n,n ˜ ∈Z
x=\frac(\text( )\!\!\pi\!\!\text( ))(\text(4))+\text( )\!\!\pi\!\!\text( ) n,n˜\in Z
We found the root, we solved the first part of the problem, that is, we honestly earned one primary point out of two.
Let's move on to the second part: find the roots of this equation that belong to the interval, or, more precisely, to the segment
[\left[ -2\text( )\!\!\pi\!\!\text( );-\frac(\text( )\!\!\pi\!\!\text( ))(2 ) \right]\]. I suggest, as in last time solve this expression graphically, i.e. draw a circle, mark the beginning in it, i.e. 0, as well as the ends of the segment:
On the segment
−2 π ;− π 2
2\text( )\!\!\pi\!\!\text( );-\frac(\pi )(2) you need to find all the values that belong to
π 4 +π n
\frac(\text( )\!\!\pi\!\!\text( ))(\text(4))+\text( )\!\!\pi\!\!\text( )n. And now the fun part: the fact is that the point itself π 4 \frac(\text( )\!\!\pi\!\!\text( ))(4) does not belong to the segment
[ −2 π ;− π 2 ] ,
\left[ -2\text( )\!\!\pi\!\!\text( );-\frac(\text( )\!\!\pi\!\!\text( ))(2) \right], this is obvious:
π 4 ∉˜ [ −2 π ;− π 2 ]
\frac(\text( )\!\!\pi\!\!\text( ))(4)\notin ˜\left[ -2\text( )\!\!\pi\!\!\text( );-\text( )\frac(\text( )\!\!\pi\!\!\text( ))(2) \right]
If only because both ends of this segment are negative, and the number π 4 \frac(\text( )\!\!\pi\!\!\text( ))(4) positive, but on the other hand, some values of the form
π 4 +π n
\frac(\text( )\!\!\pi\!\!\text( ))(4)+\text( )\!\!\pi\!\!\text( )n still belong to our segment. So how do you highlight them? Very simple: take the end of the segment
−2π
2\text( )\!\!\pi\!\!\text( ) and add π 4 \frac(\text( )\!\!\pi\!\!\text( ))(\text(4)), i.e. everything happens the same as if we started the report not from 0, and from −2π-2\text( )\!\!\pi\!\!\text() and we have the first point:
x=−2 π + π 4 =−7π 4
x=-2\text( )\!\!\pi\!\!\text( )+\frac(\text( )\!\!\pi\!\!\text( ))(4)=- \frac(7\text( )\!\!\pi\!\!\text( ))(4)
Now the second number:
x=−2 π + π 4 + π =− 3π 4
x=-2\text( )\!\!\pi\!\!\text( )+\frac(\text( )\!\!\pi\!\!\text( ))(\text(4 ))+\text( )\!\!\pi\!\!\text( )=-\frac(3\text( )\!\!\pi\!\!\text( ))(4)
This is the second meaning. There are no other roots, because we ourselves, when marking them and when marking our segment of the limitation, discovered that within this segment there are only two types - π 4 \frac(\text( )\!\!\pi\!\!\text( ))(\text(4)) and π 4 + π \frac(\text( )\!\!\pi\!\!\text( ))(4)+\text( )\!\!\pi\!\!\text( ). These points are us and ours. We write out the answer:
−7π 4 ;−3π 4
-\frac(7\text( )\!\!\pi\!\!\text( ))(4);-\frac(3\text( )\!\!\pi\!\!\text( ))(4)
For such a decision you will receive two primary scores out of two possible ones.
What you need to remember for the right decision
Once again the key steps to follow. First of all, you need to know the calculations of the double angle of a sine or cosine, in particular, in our problem, the cosine of a double angle. In addition, after using it, you need to solve the simplest trigonometric equation. The solution is quite simple, but you need to write and check that cosx\cos x in our construction is not equal to 0. After trigonometric equation we get an elementary expression, in our case it is tgx=1 tgx=1, which can be easily solved by standard formulas, known since 9-10 grade. Thus, we will solve the example and get the answer to the first part of the task - the set of all roots. In our case it is
π 4 + π n,n∈Z
\frac(\text( )\!\!\pi\!\!\text( ))(\text(4))+\text( )\!\!\pi\!\!\text( )n, n\in ˜Z. Then all that remains is to select the roots belonging to the segment
[ −2 π ;− π 2 ]
\left[ -2\text( )\!\!\pi\!\!\text( );-\frac(\text( )\!\!\pi\!\!\text( ))(2) \right]. To do this, we again draw a trigonometric circle, mark our roots and our segment on it, and then count from the end that same π 4 \frac(\text( )\!\!\pi\!\!\text( ))(4) and π 4 + π \frac(\text( )\!\!\pi\!\!\text( ))(4)+\text( )\!\!\pi\!\!\text( ), which were obtained during marking all roots of the form π 4 +π n\frac(\text( )\!\!\pi\!\!\text( ))(\text(4))+\text( )\!\!\pi\!\!\text( )n. After a simple calculation we got two specific roots, namely,
−7π 4
-\frac(7\text( )\!\!\pi\!\!\text( ))(4) and
−3π 4
-\frac(3\text( )\!\!\pi\!\!\text( ))(4), which are the answer to the second part of the problem, i.e. the roots belonging to the segment
[ −2 π ;− π 2 ]
\left[ -2\text( )\!\!\pi\!\!\text( );-\frac(\text( )\!\!\pi\!\!\text( ))(2) \right].
Key points
To easily cope with C1 problems of this type, remember two basic formulas:
- Sine of double angle:
sin2 α =2sin α cos α
\sin 2\text( )\!\!\alpha\!\!\text( )=2\sin \text( )\!\!\alpha\!\!\text( )\cos \text( )\ !\!\alpha\!\!\text( ) - this formula for sines always works in this form;
- Cosine of double angle: cos2 α =co s2 α−si n2 α \cos 2\text( )\!\!\alpha\!\!\text( =)co((s)^(2))\text( )\!\!\alpha\!\!\text( ) -si((n)^(2))\text( )\!\!\alpha\!\!\text( ) - and here there are possible options.
The first one is clear. But what options are possible in the second case? The fact is that the cosine of a double angle can be written in different ways:
cos2 α =cos2 α −sin2 α =2cos2 α −1=1−2sin2 α
\cos 2\text( )\!\!\alpha\!\!\text( )=\cos 2\text( )\!\!\alpha\!\!\text( )-\sin 2\text( )\!\!\alpha\!\!\text( )=2\cos 2\text( )\!\!\alpha\!\!\text( )-1=1-2\sin 2\text( )\!\!\alpha\!\!\text( )
These equalities follow from the basic trigonometric identity. Well, which equality to choose when solving concrete example C1? It's simple: if you plan to reduce the construction to sines, then choose the last expansion, which contains only
sin2 α
\sin 2\text( )\!\!\alpha\!\!\text( ). Conversely, if you want to reduce the entire expression to working with cosines, choose the second option - the one where cosine is the only trigonometric function.
Most Frequently Asked Questions
Is it possible to make a stamp on a document according to the sample provided? Answer Yes, it's possible. Send a scanned copy or a good quality photo to our email address, and we will make the necessary duplicate.
What types of payment do you accept?
Answer You can pay for the document upon receipt by the courier, after checking the correctness of completion and quality of execution of the diploma. This can also be done at the office of postal companies offering cash on delivery services.
All terms of delivery and payment for documents are described in the “Payment and Delivery” section. We are also ready to listen to your suggestions regarding the terms of delivery and payment for the document.
Can I be sure that after placing an order you will not disappear with my money? Answer We have quite a long experience in the field of diploma production. We have several websites that are constantly updated. Our specialists work in different parts of the country, producing over 10 documents a day. Over the years, our documents have helped many people solve employment problems or move to higher-paying jobs. We have earned trust and recognition among clients, so there is absolutely no reason for us to do this. Moreover, this is simply impossible to do physically: you pay for your order the moment you receive it in your hands, there is no prepayment.
Can I order a diploma from any university? Answer In general, yes. We have been working in this field for almost 12 years. During this time, an almost complete database of documents issued by almost all universities in the country and for different years of issue was formed. All you need is to select a university, specialty, document, and fill out the order form.
What to do if you find typos and errors in a document?
Answer When receiving a document from our courier or postal company, we recommend that you carefully check all the details. If a typo, error or inaccuracy is discovered, you have the right not to pick up the diploma, but you must indicate the detected defects personally to the courier or in writing by sending an email.
We will correct the document as soon as possible and resend it to the specified address. Of course, shipping will be paid by our company.
To avoid such misunderstandings, before filling out the original form, we email the customer a mock-up of the future document for checking and approval of the final version. Before sending the document by courier or mail, we also take additional photos and videos (including in ultraviolet light) so that you have a clear idea of what you will receive in the end.
What should I do to order a diploma from your company?
Answer To order a document (certificate, diploma, academic certificate, etc.), you must fill out the online order form on our website or provide your email so that we can send you an application form, which you need to fill out and send back to us.
If you do not know what to indicate in any field of the order form/questionnaire, leave them blank. Therefore, we will clarify all the missing information over the phone.
Latest reviews
Alexei:
I needed to acquire a diploma to get a job as a manager. And the most important thing is that I have both experience and skills, but I can’t get a job without a document. Once I came across your site, I finally decided to buy a diploma. The diploma was completed in 2 days!! Now I have a job that I never dreamed of before!! Thank you!