RD Sharma Class 12 Exercise 3.13 Inverse Trigonometric Functions Solutions Maths - Download PDF Free Online

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# RD Sharma Class 12 Exercise 3.13 Inverse Trigonometric Functions Solutions Maths - Download PDF Free Online

Edited By Kuldeep Maurya | Updated on Jan 21, 2022 11:46 AM IST

The RD Sharma solution books are the best companion for the class 12 students. When a teacher is the best guide at the school to clear the doubts, the RD Sharma solution books are the best guides for the students at their home. Many students find it challenging to cope with the concepts in the 3rd chapter of mathematics, Inverse Trigonometry. The RD Sharma Class 12th Exercise 3.13 books are a boon for them.

Also Read - RD Sharma Solution for Class 9 to 12 Maths

Inverse Trigonometry is not an easy chapter to solve, and there are numerous formulas and methods to bring the right solutions. Especially in the thirteenth exercise, it is even more challenging for students to solve when they are unclear about the previous exercises. Exercise 3.13 consists of problems that include the concepts like sine, cosine, tangent, cotangent, secant, and cosecant functions. There are five questions, including its subparts, to be solved in this exercise. To make it easier for the students, the RD Sharma Class 12 Chapter 3 Exercise 3.13 will lend a helping hand to the students.

## Inverse Trigonometry Function Ex 3.13

Inverse Trigonometric Function Exercise 3.13 Question 1

Given:
$\cos^{-1}\frac{x}{2}+\cos^{-1}\frac{y}{3}= \alpha$
To prove:
$9x^{2}-12xy\cos \alpha +4y^{2}= 36\sin ^{2}\alpha$
Hint:
We are applying the formula of
$\cos^{-1}x+\cos^{-1}y= \cos^{-1}\left [ xy-\left ( \sqrt{1-x^{2}} \right )\left ( \sqrt{1-y^{2}} \right )\right ]$
Solution:
We have
$\cos^{-1}\frac{x}{2}+\cos^{-1}\frac{y}{3}= \cos^{-1}\left [ \frac{x}{2}\times \frac{y}{3}-\left ( \sqrt{1-\left ( \frac{x}{2} \right )^{2}} \right )\left ( \sqrt{1-\left ( \frac{y}{3} \right )^{2}} \right )\right ]$
$= \cos^{-1}\left [ \frac{xy}{6}-\frac{\sqrt{4-x^{2}}}{2}\times \frac{\sqrt{9-y^{2}}}{3} \right ]$
$= \cos^{-1}\left [ \frac{xy-\sqrt{4-x^{2}}\times \sqrt{9-y^{2}}}{6} \right ]= \alpha$ (Let)
$\Rightarrow xy-\sqrt{4-x^{2}} \times \sqrt{9-y^{2}} = 6\cos \alpha$
$\Rightarrow \left ( xy-6\cos \alpha \right )= \sqrt{4-x^{2}}\times \sqrt{9-y^{2}}$
On squaring both sides, we get
$\! \! \! \! \! \! \! \! \Rightarrow \left ( xy-6\cos \alpha \right )^{2}= \left ( 4-x^{2} \right )\left ( 9-y ^{2}\right )\\\Rightarrow x^{2y^{2}}+36\cos ^{2}\alpha -12xy\cos \alpha = 36-9x^{2}-4y^{2}+x^{2}y^{2}$
$\! \! \! \! \! \! \! \! \! \Rightarrow 9x^{2}+4y^{2}-36+36\cos ^{2}\alpha-12xy\cos \alpha = 0\\\Rightarrow 9x^{2}+4y^{2}-12xy\cos \alpha-36\left ( 1-\cos ^{2}\alpha \right )= 0$
$\Rightarrow 9x^{2}+4y^{2}-12xy\cos \alpha -36\sin ^{2}\alpha = 0$
$\Rightarrow 9x^{2}+4y^{2}-12xy\cos \alpha = 36\sin ^{2}\alpha$
Hence proved

Inverse Trigonometric Function Exercise 3.13 Question 2

$x= ab$
Given:
$\cos^{-1}\frac{a}{x}-\cos^{-1}\frac{b}{x}= \cos^{-1}\frac{1}{b}-\cos^{-1}\frac{1}{a}$
Hint:
First, we will separate the angle of $a$ terms then we will apply the formula,
$\cos^{-1}x+\cos^{-1}y= \cos^{-1}\left [ xy-\left ( \sqrt{1-x^{2}} \right )\left ( \sqrt{1-y^{2}} \right ) \right ]$
$\cos^{-1}\frac{a}{x}+\cos^{-1}\frac{1}{a} = \cos^{-1}\frac{1}{b}+\cos^{-1}\frac{b}{x}$
Solution:
We know that,
$\cos^{-1}x+\cos^{-1}y= \cos^{-1}\left [ xy-\left ( \sqrt{1-x^{2}} \right )\left ( \sqrt{1-y^{2}} \right ) \right ]$
Substituting the values in the formula, we get
$\cos^{-1}\left [ \frac{1}{x}-\sqrt{1-\left ( \frac{a}{x} \right )^{2}} \sqrt{1-\left ( \frac{1}{a} \right )^{2}}\right ]= \cos^{-1}\left [ \frac{1}{x}\sqrt{1-\left ( \frac{b}{x} \right )^{2}}\sqrt{1-\left ( \frac{1}{b} \right )^{2}} \right ]$
$\Rightarrow \frac{1}{x}-\sqrt{1-\left ( \frac{a}{x} \right )^{2}} \sqrt{1-\left ( \frac{1}{a} \right )^{2}}=\frac{1}{x}-\sqrt{1-\left ( \frac{b}{x} \right )^{2}}\sqrt{1-\left ( \frac{1}{b} \right )^{2}}$
$\Rightarrow\sqrt{1-\left ( \frac{a}{x} \right )^{2}} \sqrt{1-\left ( \frac{1}{a} \right )^{2}}=\sqrt{1-\left ( \frac{b}{x} \right )^{2}}\sqrt{1-\left ( \frac{1}{b} \right )^{2}}$
Squaring on both sides, we get
$\Rightarrow \left ( 1-\left ( \frac{a}{x} \right )^{2}\right )\left ( 1-\left ( \frac{1}{a} \right )^{2} \right )=\left ( 1-\left ( \frac{b}{x} \right ) ^{2}\right )\left ( 1-\left ( \frac{1}{b}\right )^{2} \right )$
$\Rightarrow 1-\left ( \frac{a}{x} \right )^{2}-\left ( \frac{1}{a} \right )^{2}+\left ( \frac{1}{x} \right )^{2}= 1-\left ( \frac{b}{x} \right )^{2}-\left ( \frac{1}{b} \right )^{2}+\left ( \frac{1}{x} \right )^{2}$
$\Rightarrow \left ( \frac{b}{x} \right )^{2}-\left ( \frac{a}{x} \right )^{2}= \left ( \frac{1}{a} \right )^{2}-\left ( \frac{1}{b} \right )^{2}$
On Simplifying, we get
$\Rightarrow \left ( b^{2} -a^{2}\right )a^{2}b^{2}= x^{2}\left ( b^{2}-a^{2} \right )$
$\Rightarrow x^{2}= a^{2}b^{2}$
$\Rightarrow x= ab$
Hence the result.

Inverse Trigonometric Function Exercise 3.13 Question 3

$x= \frac{1}{2}$
Given:
$\cos^{-1}\sqrt{3x}+\cos^{-1}x= \frac{x}{2}$
Hint:
We are applying the formula,
$\cos^{-1}x+\cos^{-1}y= \cos^{-1}\left [ xy-\left ( \sqrt{1-x^{2}} \right )\left ( \sqrt{1-y^{2}} \right ) \right ]$
Solution:
We have
$\cos^{-1}\sqrt{3x}+\cos^{-1}x= \frac{x}{2}$
Using formula written on the hint, we get
$\Rightarrow \cos^{-1}\left [ \sqrt{3x}\times x-\left ( \sqrt{1-\left ( \sqrt{3x} \right )^{2}} \right )\left ( \sqrt{1-x^{2}} \right ) \right ]= \frac{x}{2}$
$\Rightarrow \cos^{-1}\left [ \sqrt{3x^{2}}-\left ( \sqrt{1-3x ^{2}} \right )\left ( \sqrt{1-x^{2}} \right ) \right ]= \frac{x}{2}$
$\Rightarrow \sqrt{3x^{2}}-\left ( \sqrt{1-3x ^{2}} \right )\left ( \sqrt{1-x^{2}} \right )= \cos \frac{x}{2}$
$\Rightarrow \sqrt{3x^{2}}-\left ( \sqrt{1-3x ^{2}} \right )\left ( \sqrt{1-x^{2}} \right )= 0\; \; \; \; \; \; \left [ \because \cos \frac{x}{2}= 0 \right ]$
$\Rightarrow \sqrt{3x^{2}}-\left ( \sqrt{1-3x ^{2}} \right )\left ( \sqrt{1-x^{2}} \right )$
Squaring on both sides, we get
$\Rightarrow 3x^{4}-\left ( 1-3x ^{2} \right )\left ( 1-x^{2} \right )$
$\Rightarrow 3x^{4}= 1-x ^{2} -3x^{2}+3x^{4}$
$\Rightarrow 1-4x^{2}= 0$
$\Rightarrow 4x^{2}= 1$
$\Rightarrow x^{2}= \frac{1}{4}$
$\Rightarrow x=\pm \frac{1}{2}\; \; \; \; \; \; \; \; \; [ As -\frac{1}{2} is\: not\: satisf\! ying \: the\: equation]$
Hence the result is $x= \frac{1}{2}$

Inverse Trigonometric Function Exercise 3.13 Question 4

Given:
$\cos^{-1}\left ( \frac{4}{5} \right )+\cos^{-1}\left ( \frac{12}{13} \right )$
To prove:
$\cos^{-1}\left ( \frac{4}{5} \right )+\cos^{-1}\left ( \frac{12}{13} \right )= \cos^{-1}\left ( \frac{33}{65} \right )$
Hint:
We will use the formula on L.H.S
$\cos^{-1}x+\cos^{-1}y= \cos^{-1}\left [ xy-\left ( \sqrt{1-x^{2}} \right )\left ( \sqrt{1-y^{2}} \right ) \right ]$
Solution:
Taking L.H.S
L.H.S
$\cos^{-1}\left ( \frac{4}{5} \right )+\cos^{-1}\left ( \frac{12}{13} \right )$

$\Rightarrow \cos^{-1}\left [ \frac{48}{65}-\sqrt{1-\frac{16}{25} }\sqrt{1-\frac{144}{169}}\right ]$
$\Rightarrow \cos^{-1}\left [ \frac{48}{65}-\frac{3}{5}\times \frac{5}{13}\right ]$
$\Rightarrow \cos^{-1}\left ( \frac{33}{65} \right )$
$=$ R.H.S
Hence we get R.H.S.

Inverse Trigonometric Function Exercise 3.13 Question 5

Given:
$\cos^{-1}\frac{12}{13}+\sin^{-1}\frac{3}{5}$
Hint:
First, we convert $\cos^{-1}$ to $\sin^{-1}$ and then use $\sin \left ( a+b \right )$ formula.
Solution:
Let $a= \cos^{-1}\frac{12}{13}$ and $b= \sin^{-1}\frac{3}{5}$
Finding $\sin a , \cos a$.
Let $a= \cos^{-1}\frac{12}{13}$
$\cos a= \left (\frac{12}{13} \right )$
We know that
$\sin a= \sqrt{1-\cos ^{2}}a$
$= \sqrt{1-\left ( \frac{12}{13} \right )^{2}}$
$= \sqrt{\frac{25}{169}}$
$= \frac{5}{13}$
Again, finding $\sin b, \cos b$
Let $b= \sin^{-1}\left ( \frac{3}{5} \right )$
$\Rightarrow \sin b= \frac{3}{5}$
We know that,
$\cos b= \sqrt{1-\sin ^{2}b}$
$\Rightarrow \cos b= \sqrt{1-\left ( \frac{3}{5} \right )^{2}}$
$= \sqrt{\frac{16}{25}}= \frac{4}{5}$
$\cos b= \frac{4}{5}$
We know that,
$\sin \left ( a+b \right )= \sin a\cos b+\cos a\sin b$
Putting, $\sin a= \frac{5}{13},\cos a= \frac{12}{13}$
$\sin b= \frac{3}{5},\cos b= \frac{4}{5}$
We have,
$\sin \left ( a+b \right )= \frac{5}{13}\times \frac{4}{5}+\frac{12}{13}\times \frac{3}{5}$
$= \frac{20}{65}+\frac{36}{65}$
$\sin \left ( a+b \right )= \frac{56}{65}$
$a+b =\sin^{-1} \frac{56}{65}$
Putting the value of $a,b$
$\cos^{-1}\frac{12}{13}+\sin^{-1}\frac{3}{5}= \sin^{-1}\frac{56}{65}$
LHS=RHS
Hence proved.

The hesitations regarding the accuracy of the answers need not prevail. Every solution is provided by the best team of teachers and professors who are experts in the educational field. They have also followed the NCERT pattern to make it easy for the CBSE board students to follow. You will find the Inverse Trigonometry chapter very easy to solve once you start using the Class 12 RD Sharma Chapter 3 Exercise 3.13 Solution book.

This material gives you the freedom to choose the method you want to adapt in finding answers. This is due to the presence of elaborated solutions as well as shortcuts techniques given in the book. Moreover, the students can download the RD Sharma Class 12 Solutions Inverse Trigonometry Ex 3.13 from top educational websites like Career 360. And the bonus surprise is that you need not pay even a single rupee to access this solution book. Everything is available free of cost for the benefit of the students.

The wide usage of the RD Sharma solutions book and its best solutions has led to its strong recognition. Hence, there are many chances that your class 12 public mathematics exam will have questions from this book. Therefore, the RD Sharma Class 12 Solutions Chapter 3 ex 3.13 material is suitable for your exam preparations. Make it a practice by using this book for solving your homework and assignments. Download your copy of the RD Sharma solutions book from the Career360 website and start to solve the sums accordingly.

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1. What book should I refer to to clear my doubts on Chapter 3, Inverse Trigonometry?

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2. Which is the best guide for class 12 students to prepare well for their public exam?

Experts in the educational sector provide the solutions in the RD Sharma books. The students can use it to prepare for their exams, homework, and assignments.

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rrespective of how they perform in exams, every student can use the RD Sharma solutions book to learn the concepts. The solutions are given in various methods; hence the students can prefer the best way they find easy and go ahead with it.

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The career as a Talent Agent is filled with responsibilities. A Talent Agent is someone who is involved in the pre-production process of the film. It is a very busy job for a Talent Agent but as and when an individual gains experience and progresses in the career he or she can have people assisting him or her in work. Depending on one’s responsibilities, number of clients and experience he or she may also have to lead a team and work with juniors under him or her in a talent agency. In order to know more about the job of a talent agent continue reading the article.

If you want to know more about talent agent meaning, how to become a Talent Agent, or Talent Agent job description then continue reading this article.

3 Jobs Available

Radio Jockey is an exciting, promising career and a great challenge for music lovers. If you are really interested in a career as radio jockey, then it is very important for an RJ to have an automatic, fun, and friendly personality. If you want to get a job done in this field, a strong command of the language and a good voice are always good things. Apart from this, in order to be a good radio jockey, you will also listen to good radio jockeys so that you can understand their style and later make your own by practicing.

A career as radio jockey has a lot to offer to deserving candidates. If you want to know more about a career as radio jockey, and how to become a radio jockey then continue reading the article.

3 Jobs Available
##### Videographer

Careers in videography are art that can be defined as a creative and interpretive process that culminates in the authorship of an original work of art rather than a simple recording of a simple event. It would be wrong to portrait it as a subcategory of photography, rather photography is one of the crafts used in videographer jobs in addition to technical skills like organization, management, interpretation, and image-manipulation techniques. Students pursue Visual Media, Film, Television, Digital Video Production to opt for a videographer career path. The visual impacts of a film are driven by the creative decisions taken in videography jobs. Individuals who opt for a career as a videographer are involved in the entire lifecycle of a film and production.

2 Jobs Available
##### Multimedia Specialist

A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications.

2 Jobs Available
##### Visual Communication Designer

Individuals who want to opt for a career as a Visual Communication Designer will work in the graphic design and arts industry. Every sector in the modern age is using visuals to connect with people, clients, or customers. This career involves art and technology and candidates who want to pursue their career as visual communication designer has a great scope of career opportunity.

2 Jobs Available
##### Copy Writer

In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook.

5 Jobs Available
##### Journalist

Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.

3 Jobs Available
##### Publisher

For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.

3 Jobs Available
##### Vlogger

In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion. Ever since internet cost got reduced the viewership for these types of content has increased on a large scale. Therefore, the career as vlogger has a lot to offer. If you want to know more about the career as vlogger, how to become a vlogger, so on and so forth then continue reading the article. Students can visit Jamia Millia Islamia, Asian College of Journalism, Indian Institute of Mass Communication to pursue journalism degrees.

3 Jobs Available
##### Editor

Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.

3 Jobs Available
##### Content Writer

Content writing is meant to speak directly with a particular audience, such as customers, potential customers, investors, employees, or other stakeholders. The main aim of professional content writers is to speak to their targeted audience and if it is not then it is not doing its job. There are numerous kinds of the content present on the website and each is different based on the service or the product it is used for.

2 Jobs Available
##### Reporter

Individuals who opt for a career as a reporter may often be at work on national holidays and festivities. He or she pitches various story ideas and covers news stories in risky situations. Students can pursue a BMC (Bachelor of Mass Communication), B.M.M. (Bachelor of Mass Media), or MAJMC (MA in Journalism and Mass Communication) to become a reporter. While we sit at home reporters travel to locations to collect information that carries a news value.

2 Jobs Available
##### Linguist

Linguistic meaning is related to language or Linguistics which is the study of languages. A career as a linguistic meaning, a profession that is based on the scientific study of language, and it's a very broad field with many specialities. Famous linguists work in academia, researching and teaching different areas of language, such as phonetics (sounds), syntax (word order) and semantics (meaning).

Other researchers focus on specialities like computational linguistics, which seeks to better match human and computer language capacities, or applied linguistics, which is concerned with improving language education. Still, others work as language experts for the government, advertising companies, dictionary publishers and various other private enterprises. Some might work from home as freelance linguists. Philologist, phonologist, and dialectician are some of Linguist synonym. Linguists can study French, German, Italian

2 Jobs Available
##### Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.

3 Jobs Available
##### Quality Controller

A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.

A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.

3 Jobs Available
##### Production Manager

Production Manager Job Description: A Production Manager is responsible for ensuring smooth running of manufacturing processes in an efficient manner. He or she plans and organises production schedules. The role of Production Manager involves estimation, negotiation on budget and timescales with the clients and managers.

3 Jobs Available

A Team Leader is a professional responsible for guiding, monitoring and leading the entire group. He or she is responsible for motivating team members by providing a pleasant work environment to them and inspiring positive communication. A Team Leader contributes to the achievement of the organisation’s goals. He or she improves the confidence, product knowledge and communication skills of the team members and empowers them.

2 Jobs Available
##### Quality Systems Manager

A Quality Systems Manager is a professional responsible for developing strategies, processes, policies, standards and systems concerning the company as well as operations of its supply chain. It includes auditing to ensure compliance. It could also be carried out by a third party.

2 Jobs Available
##### Merchandiser

A career as a merchandiser requires one to promote specific products and services of one or different brands, to increase the in-house sales of the store. Merchandising job focuses on enticing the customers to enter the store and hence increasing their chances of buying a product. Although the buyer is the one who selects the lines, it all depends on the merchandiser on how much money a buyer will spend, how many lines will be purchased, and what will be the quantity of those lines. In a career as merchandiser, one is required to closely work with the display staff in order to decide in what way a product would be displayed so that sales can be maximised. In small brands or local retail stores, a merchandiser is responsible for both merchandising and buying.

2 Jobs Available
##### Procurement Manager

The procurement Manager is also known as  Purchasing Manager. The role of the Procurement Manager is to source products and services for a company. A Procurement Manager is involved in developing a purchasing strategy, including the company's budget and the supplies as well as the vendors who can provide goods and services to the company. His or her ultimate goal is to bring the right products or services at the right time with cost-effectiveness.

2 Jobs Available
##### Production Planner

Individuals who opt for a career as a production planner are professionals who are responsible for ensuring goods manufactured by the employing company are cost-effective and meets quality specifications including ensuring the availability of ready to distribute stock in a timely fashion manner.

2 Jobs Available
##### Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.

3 Jobs Available
##### ITSM Manager

ITSM Manager is a professional responsible for heading the ITSM (Information Technology Service Management) or (Information Technology Infrastructure Library) processes. He or she ensures that operation management provides appropriate resource levels for problem resolutions. The ITSM Manager oversees the level of prioritisation for the problems, critical incidents, planned as well as proactive tasks.

3 Jobs Available
##### Information Security Manager

Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack

3 Jobs Available
##### Computer Programmer

Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.

3 Jobs Available
##### Computer System Analyst

Individuals in the computer systems analyst career path study the hardware and applications that are part of an organization's computer systems, as well as how they are used. They collaborate closely with managers and end-users to identify system specifications and business priorities, as well as to assess the efficiency of computer systems and create techniques to boost IT efficiency. Individuals who opt for a career as a computer system analyst support the implementation, modification, and debugging of new systems after they have been installed.

2 Jobs Available
##### Test Manager

A Test Manager is a professional responsible for planning, coordinating and controlling test activities. He or she develops test processes and strategies to analyse and determine test methods and tools for test activities. The test manager jobs involve documenting tests that have been carried out, analysing and evaluating software quality to determine further recommended procedures.

2 Jobs Available
##### Azure Developer

A career as Azure Developer comes with the responsibility of designing and developing cloud-based applications and maintaining software components. He or she possesses an in-depth knowledge of cloud computing and Azure app service.

2 Jobs Available
##### Deep Learning Engineer

A Deep Learning Engineer is an IT professional who is responsible for developing and managing data pipelines. He or she is knowledgeable about analyzing and storing data collected from various sources.  A Career as a Deep Learning Engineer needs to help the  data scientists and analysts to create effective data sets.

2 Jobs Available