# The trigonometric identities discussion paper

Trigonometry is a branch of Mathematics dealing with the angle and the sides of right-angled triangles. There are several mathematical formulas in trigonometry which are generally divided into two parts: trigonometric identity or trigonometric ratios. Trigonometric identities are the equations based on the laws of triangle, they relate the side and the angle of the triangle. These are the fundamental identities which will be used for solving trigonometric problems in various applications.

There are basically six trigonometric functions; sine, cosine, tangent, cosecant, secant, and cotangent. These identities relate all these functions with each other and deduce several other relationships with different operations of the angle of the triangle. There are many identities but, in this section, these are divided into two parts: one consists of important identities which one must remember. They are more commonly used for solving trigonometric problems and the other one is second string trigonometric identities. These are not common as they have lesser number of applications and can be used only when needed.

Don't use plagiarized sources. Get Your Custom Essay on
The trigonometric identities discussion paper
Just from \$9/Page

Important trigonometric identities
These are very basic and important trigonometric identities. These identities have several applications.

Defining relationship between various trigonometric functions.

\begin{array}{ll}\tan \theta=\frac{\sin \theta}{\cos \theta} & \sec \theta=\frac{1}{\cos \theta} \\ \cot \theta=\frac{1}{\tan \theta}=\frac{\cos \theta}{\sin \theta} & \csc \theta=\frac{1}{\sin \theta}\end{array}
tanÎ¸=
cosÎ¸
sinÎ¸
â€‹

cotÎ¸=
tanÎ¸
1
â€‹
=
sinÎ¸
cosÎ¸
â€‹

â€‹

secÎ¸=
cosÎ¸
1
â€‹

cscÎ¸=
sinÎ¸
1
â€‹

â€‹

Pythagorean identities: These are one of the most important identities derived by the Pythagoras theorem of the right-angled triangle.

\sin ^{2} \theta+\cos ^{2} \theta=1sin
2
Î¸+cos
2
Î¸=1
\tan ^{2} \theta+1=\sec ^{2} \thetatan
2
Î¸+1=sec
2
Î¸
1+\cot ^{2} \theta=\csc ^{2} \theta1+cot
2
Î¸=csc
2
Î¸

Complements identities: These identities represent the trigonometric functions into their co-functions by using complementary angle.

\sin \theta=\cos \left(\frac{\pi}{2}-\theta\right)sinÎ¸=cos(
2
Ï€
â€‹
âˆ’Î¸)
\tan \theta=\cot \left(\frac{\pi}{2}-\theta\right)tanÎ¸=cot(
2
Ï€
â€‹
âˆ’Î¸)
\sec \theta=\csc \left(\frac{\pi}{2}-\theta\right)secÎ¸=csc(
2
Ï€
â€‹
âˆ’Î¸)

Periodicity identity: In these identities; cosine, sine, cosecant, and secant are having 2pi while tangent and cotangent are having a period of pi.

\sin (\theta+2 \pi)=\sin \thetasin(Î¸+2Ï€)=sinÎ¸
\cos (\theta+2 \pi)=\cos \thetacos(Î¸+2Ï€)=cosÎ¸
\tan (\theta+\pi)=\tan \thetatan(Î¸+Ï€)=tanÎ¸

Negative angle identities: We know that sine, cosecant, cotangent, and tangent are all odd functions whereas secant and cosine are both even functions.

\sin (-\theta)=-\sin \thetasin(âˆ’Î¸)=âˆ’sinÎ¸
\cos (-\theta)=\cos \thetacos(âˆ’Î¸)=cosÎ¸
\tan (-\theta)=-\tan \thetatan(âˆ’Î¸)=âˆ’tanÎ¸

Sum and difference identities: These identities are also called Ptolemyâ€™s identities which provide the sum and difference formulas for trigonometric functions.

\sin (\alpha+\beta)=\sin \alpha \cos \beta+\cos \alpha \sin \betasin(Î±+Î²)=sinÎ±cosÎ²+cosÎ±sinÎ²
\cos (\alpha+\beta)=\cos \alpha \cos \beta-\sin \alpha \sin \betacos(Î±+Î²)=cosÎ±cosÎ²âˆ’sinÎ±sinÎ²
\tan (\alpha+\beta)=\frac{\tan \alpha+\tan \beta}{1-\tan \alpha \cdot \tan \beta}tan(Î±+Î²)=
1âˆ’tanÎ±â‹…tanÎ²
tanÎ±+tanÎ²
â€‹

\sin (\alpha-\beta)=\sin \alpha \cos \beta-\cos \alpha \sin \betasin(Î±âˆ’Î²)=sinÎ±cosÎ²âˆ’cosÎ±sinÎ²
\cos (\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \betacos(Î±âˆ’Î²)=cosÎ±cosÎ²+sinÎ±sinÎ²
\tan (\alpha-\beta)=\frac{\tan \alpha-\tan \beta}{1+\tan \alpha \cdot \tan \beta}tan(Î±âˆ’Î²)=
1+tanÎ±â‹…tanÎ²
tanÎ±âˆ’tanÎ²
â€‹

Double angle identities: These identities provide trigonometric formulas for double angle of the triangle. For cosine, there are three formulas which can be derived from each other using Pythagorean identities.

\sin (2\theta ) = 2\sin \theta \cos \thetasin(2Î¸)=2sinÎ¸cosÎ¸
\cos (2\theta ) = {\cos ^2}\theta – {\sin ^2}\theta = 2{\cos ^2}\theta – 1 = 1 – 2{\sin ^2}\thetacos(2Î¸)=cos
2
Î¸âˆ’sin
2
Î¸=2cos
2
Î¸âˆ’1=1âˆ’2sin
2
Î¸
\tan (2\theta ) = \frac{{2\tan \theta }}{{1 – {{\tan }^2}\theta }}tan(2Î¸)=
1âˆ’tan
2
Î¸
2tanÎ¸
â€‹

Second string trigonometric identities
These identities are not so common and only come into consideration when in use. But these identities have much importance in solving the complex equations. Although these identities have lesser number of applications, but when needed can be put directly which will ease the solution.

Supplementary identities: These identities provide the trigonometric formulas for supplementary angles (pi-Î¸) of the triangle.

\sin (\pi – \theta ) = \sin \thetasin(Ï€âˆ’Î¸)=sinÎ¸
\cos (\pi – \theta ) = – \cos \thetacos(Ï€âˆ’Î¸)=âˆ’cosÎ¸
\tan (\pi – \theta ) = – \tan \thetatan(Ï€âˆ’Î¸)=âˆ’tanÎ¸

Half angle identities: In these identities, the negative and positive sign can be taken depending upon the quadrant of angle Î¸/2 i.e., if it is an acute angle then positive value will be taken.

\sin \frac{\theta}{2}=\pm \sqrt{\frac{1-\cos \theta}{2}}sin
2
Î¸
â€‹
=Â±
2
1âˆ’cosÎ¸
â€‹

â€‹

\cos \frac{\theta}{2}=\pm \sqrt{\frac{1+\cos \theta}{2}}cos
2
Î¸
â€‹
=Â±
2
1+cosÎ¸
â€‹

â€‹

\tan \frac{\theta}{2}=\frac{\sin \theta}{1+\cos \theta}=\frac{1-\cos \theta}{\sin \theta}tan
2
Î¸
â€‹
=
1+cosÎ¸
sinÎ¸
â€‹
=
sinÎ¸
1âˆ’cosÎ¸
â€‹

Product sum identities: These identities are useful if one wants to change sine and cosines sum and difference into their products.

\sin \alpha+\sin \beta=2 \sin \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}sinÎ±+sinÎ²=2sin
2
Î±+Î²
â€‹
cos
2
Î±âˆ’Î²
â€‹

\cos \alpha+\cos \beta=2 \cos \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}cosÎ±+cosÎ²=2cos
2
Î±+Î²
â€‹
cos
2
Î±âˆ’Î²
â€‹

\sin \alpha-\sin \beta=2 \cos \frac{\alpha+\beta}{2} \sin \frac{\alpha-\beta}{2}sinÎ±âˆ’sinÎ²=2cos
2
Î±+Î²
â€‹
sin
2
Î±âˆ’Î²
â€‹

\cos \alpha+\cos \beta=2 \cos \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}cosÎ±+cosÎ²=2cos
2
Î±+Î²
â€‹
cos
2
Î±âˆ’Î²
â€‹

Product identities: These identities provide the multiplication of sine and cosine functions with each other or with themselves.

\sin \alpha \cos \beta=\frac{\sin (\alpha+\beta)+\sin (\alpha-\beta)}{2}sinÎ±cosÎ²=
2
sin(Î±+Î²)+sin(Î±âˆ’Î²)
â€‹

\cos \alpha \cos \beta=\frac{\cos (\alpha+\beta)+\cos (\alpha-\beta)}{2}cosÎ±cosÎ²=
2
cos(Î±+Î²)+cos(Î±âˆ’Î²)
â€‹

\sin \alpha \sin \beta=\frac{\cos (\alpha+\beta)-\cos (\alpha-\beta)}{2}sinÎ±sinÎ²=
2
cos(Î±+Î²)âˆ’cos(Î±âˆ’Î²)
â€‹

Triple angle identities: If the angle of the triangle gets tripled in which respect the trigonometric functions are being calculated, then these identities can be used directly for problem solving.

\sin (3 \theta)=3 \sin \theta-4 \sin ^{3} \thetasin(3Î¸)=3sinÎ¸âˆ’4sin
3
Î¸
\cos (3 \theta)=4 \cos ^{3} \theta-\cos \thetacos(3Î¸)=4cos
3
Î¸âˆ’cosÎ¸
\tan (3 \theta)=\frac{3 \tan \theta-\tan ^{3} \theta}{1-3 \tan ^{2} \theta}tan(3Î¸)=
1âˆ’3tan
2
Î¸
3tanÎ¸âˆ’tan
3
Î¸
â€‹

## Get Professional Assignment Help Cheaply Are you busy and do not have time to handle your assignment? Are you scared that your paper will not make the grade? Do you have responsibilities that may hinder you from turning in your assignment on time? Are you tired and can barely handle your assignment? Are your grades inconsistent?

Whichever your reason is, it is valid! You can get professional academic help from our service at affordable rates. We have a team of professional academic writers who can handle all your assignments.

## Why Choose Our Academic Writing Service?

• Plagiarism free papers
• Timely delivery
• Skilled, Experienced Native English Writers
• Ability to tackle bulk assignments
• Reasonable prices

## Online Academic Help With Different Subjects

### Literature

Students barely have time to read. We got you! Have your literature essay or book review written without having the hassle of reading the book. You can get your literature paper custom-written for you by our literature specialists.

### Finance

Do you struggle with finance? No need to torture yourself if finance is not your cup of tea. You can order your finance paper from our academic writing service and get 100% original work from competent finance experts.

### Computer science

Computer science is a tough subject. Fortunately, our computer science experts are up to the match. No need to stress and have sleepless nights. Our academic writers will tackle all your computer science assignments and deliver them on time. Let us handle all your python, java, ruby, JavaScript, php , C+ assignments!

### Psychology

While psychology may be an interesting subject, you may lack sufficient time to handle your assignments. Don’t despair; by using our academic writing service, you can be assured of perfect grades. Moreover, your grades will be consistent.

### Engineering

Engineering is quite a demanding subject. Students face a lot of pressure and barely have enough time to do what they love to do. Our academic writing service got you covered! Our engineering specialists follow the paper instructions and ensure timely delivery of the paper.

### Nursing

In the nursing course, you may have difficulties with literature reviews, annotated bibliographies, critical essays, and other assignments. Our nursing assignment writers will offer you professional nursing paper help at low prices.

### Sociology

Truth be told, sociology papers can be quite exhausting. Our academic writing service relieves you of fatigue, pressure, and stress. You can relax and have peace of mind as our academic writers handle your sociology assignment.

We take pride in having some of the best business writers in the industry. Our business writers have a lot of experience in the field. They are reliable, and you can be assured of a high-grade paper. They are able to handle business papers of any subject, length, deadline, and difficulty!

### Statistics

We boast of having some of the most experienced statistics experts in the industry. Our statistics experts have diverse skills, expertise, and knowledge to handle any kind of assignment. They have access to all kinds of software to get your assignment done.

### Law

Writing a law essay may prove to be an insurmountable obstacle, especially when you need to know the peculiarities of the legislative framework. Take advantage of our top-notch law specialists and get superb grades and 100% satisfaction.

## What discipline/subjects do you deal in?

We have highlighted some of the most popular subjects we handle above. Those are just a tip of the iceberg. We deal in all academic disciplines since our writers are as diverse. They have been drawn from across all disciplines, and orders are assigned to those writers believed to be the best in the field. In a nutshell, there is no task we cannot handle; all you need to do is place your order with us. As long as your instructions are clear, just trust we shall deliver irrespective of the discipline.

## Are your writers competent enough to handle my paper?

Our essay writers are graduates with bachelor's, masters, Ph.D., and doctorate degrees in various subjects. The minimum requirement to be an essay writer with our essay writing service is to have a college degree. All our academic writers have a minimum of two years of academic writing. We have a stringent recruitment process to ensure that we get only the most competent essay writers in the industry. We also ensure that the writers are handsomely compensated for their value. The majority of our writers are native English speakers. As such, the fluency of language and grammar is impeccable.

## What if I don’t like the paper?

There is a very low likelihood that you won’t like the paper.

### Reasons being:

• When assigning your order, we match the paper’s discipline with the writer’s field/specialization. Since all our writers are graduates, we match the paper’s subject with the field the writer studied. For instance, if it’s a nursing paper, only a nursing graduate and writer will handle it. Furthermore, all our writers have academic writing experience and top-notch research skills.
• We have a quality assurance that reviews the paper before it gets to you. As such, we ensure that you get a paper that meets the required standard and will most definitely make the grade.

### In the event that you don’t like your paper:

• The writer will revise the paper up to your pleasing. You have unlimited revisions. You simply need to highlight what specifically you don’t like about the paper, and the writer will make the amendments. The paper will be revised until you are satisfied. Revisions are free of charge
• We will have a different writer write the paper from scratch.
• Last resort, if the above does not work, we will refund your money.

## Will the professor find out I didn’t write the paper myself?

Not at all. All papers are written from scratch. There is no way your tutor or instructor will realize that you did not write the paper yourself. In fact, we recommend using our assignment help services for consistent results.

## What if the paper is plagiarized?

We check all papers for plagiarism before we submit them. We use powerful plagiarism checking software such as SafeAssign, LopesWrite, and Turnitin. We also upload the plagiarism report so that you can review it. We understand that plagiarism is academic suicide. We would not take the risk of submitting plagiarized work and jeopardize your academic journey. Furthermore, we do not sell or use prewritten papers, and each paper is written from scratch.

## When will I get my paper?

You determine when you get the paper by setting the deadline when placing the order. All papers are delivered within the deadline. We are well aware that we operate in a time-sensitive industry. As such, we have laid out strategies to ensure that the client receives the paper on time and they never miss the deadline. We understand that papers that are submitted late have some points deducted. We do not want you to miss any points due to late submission. We work on beating deadlines by huge margins in order to ensure that you have ample time to review the paper before you submit it.

We have a privacy and confidentiality policy that guides our work. We NEVER share any customer information with third parties. Noone will ever know that you used our assignment help services. It’s only between you and us. We are bound by our policies to protect the customer’s identity and information. All your information, such as your names, phone number, email, order information, and so on, are protected. We have robust security systems that ensure that your data is protected. Hacking our systems is close to impossible, and it has never happened.

## How our Assignment  Help Service Works

#### 1.      Place an order

You fill all the paper instructions in the order form. Make sure you include all the helpful materials so that our academic writers can deliver the perfect paper. It will also help to eliminate unnecessary revisions.

#### 2.      Pay for the order

Proceed to pay for the paper so that it can be assigned to one of our expert academic writers. The paper subject is matched with the writer’s area of specialization.

#### 3.      Track the progress

You communicate with the writer and know about the progress of the paper. The client can ask the writer for drafts of the paper. The client can upload extra material and include additional instructions from the lecturer. Receive a paper.

The paper is sent to your email and uploaded to your personal account. You also get a plagiarism report attached to your paper.  PLACE THIS ORDER OR A SIMILAR ORDER WITH US TODAY AND GET A PERFECT SCORE!!! Published
Why we are the best
Our Free Services Why Hire Our Writers?

#### Why Choose Us? #### Why Hire Our Writers? #### How to place an order:

1. Fill the order form
2. Pay for the order
3. Track progress of your paper