Trigonometric Identity Involving Sine And Cosine

Concept Explanation

Trigonometric Identity Involving Sine And Cosine

Trigonometric Identities: The trigonometric identities are equalities which are true for every value appearing on both sides of the equal sign An equation involving trigonometric ratios of an angle is called a trigonometric identity, if it is true for all values of the angle(s) involved. These identities are useful whenever expressions involving trigonometric ratios are to be simplified.

Trigonometric Identity Involving Sine and Cosine: The identity can be stated as

$sin^2;theta + cos^2;theta = 1$

Let us try to prove these identities and use it further to simplify the various trigonometric expressions

In $large Delta ABC$ , right -angled at B according to the pythagoras theorem we have

$AB^{2}+BC^{2}=AC^{2}$ (1)

Proof ForThe Identity: Dividing each term of (1) by $AC^{2}$ , we get

$frac{AB^{2}}{AC^{2}}+frac{BC^{2}}{AC^{2}}=frac{AC^{2}}{AC^{2}}$

i.e., $left (frac{AB}{AC} right )^{2}+left (frac{BC}{AC} right)^{2}=left ( frac{AC}{AC} right)^{2}$

i.e., $(cos A)^{2}+(sin A)^{2}=1$

i.e., $cos^{2}A+sin^{2}A=1$

This is true for all A such that $0^{circ}leq Aleq 90^{circ}$ . So, this is a trigonometric identity.

Illustration: Simplify the expression:

$frac{sin Theta -2 sin^3Theta }{2 cos^3Theta -cos Theta }$

Solution: We will simplify the expression using trigonometric Identites

$frac{sin Theta -2 sin^3Theta }{2 cos ^3Theta -cos Theta }$ = $frac{sin Theta (1-2 sin^2Theta )}{cos Theta (2 cos ^2Theta -1)}$

$=tan;Theta left (frac{1-2(1-cos^2Theta )}{2 cos^2Theta -1} right)$

$=tan;Theta left (frac{1-2 +2cos^2Theta )}{2 cos^2Theta -1} right)$

$=tan Theta left (frac{2 cos^2Theta -1}{2 cos^2Theta -1} right)$

$=tan Theta$

Sample Questions

(More Questions for each concept available in Login)

Question : 1

The value of the following expression is $left ( sin ^27frac{1}{2}^0+cos^27frac{1}{2}^0 right )-left ( sin^230^0+cos^230^0 right )+left ( sin ^27^0+sin^283^0 right )$
A. 3	B. $3frac{1}{2}$	C. 2	D. 1
Right Option : D
View Explanation

Question : 2

Find the value of the expression $left ( sin^27frac{1}{2}^0+cos^27frac{1}{2}^0 right )+left ( sin^260^0+cos^260^0 right )+left ( sin ^217^0+sin^273^0 right )$
A. 3	B. $3frac{1}{2}$	C. 2	D. 3
Right Option : D
View Explanation

Question : 3

$cos^2;75^0+cos^2;15^0=$ ___________________
A. 0	B. 1	C. 2	D. $frac{1}{4}$
Right Option : B
View Explanation

Chapters

Cube and Cube Roots

Sequences and Series

Permutation and Combination

Partnership and Mixture

Introduction to Trigonometry

Probability

Surds And Indices

Problems On Boats And Stream

Rules For Divisibility

Square and Square Root

Percentages

Number System

Data Interpretation

Time, Speed and Distance

Averages Foundation

Ratio and Proportion

Time and Work

Surface Area and Volume

Simple and Compound Interest

Profit Loss and Discount

Content / Category

IMO-Mathematics Olympiad

Central Government Service

ISO - Science Olympiads

State Government Services

Banking And Insurance

Railways & Metro Services

MBA Entrance

Law Entrance

English Proficiency

Reasoning Proficiency

Computer Proficiency

Navodaya Entrance 6th & 9th

Sainik School Entrance

Andhra Bank

Class / Course

General Awareness For SSC

SSC CHSL

General Knowledge For SSC

SSC MTS

SSC CHSL Test Series

SSC CGL

Students / Parents Reviews [20]

My experience was very good with Abhyas academy. I am studying here from 6th class and I am satisfied by its results in my life. I improved a lot here ahead of school syllabus.

Ayan Ghosh

8th

My experience with Abhyas academy is very nice or it can be said wonderful. I have been studying here from seven class. I have been completing my journey of three years. I am tinking that I should ...

Hridey Preet

9th

Abhyas is good institution and a innovative institute also. It is a good platform of beginners.Due to Abhyas,he has got knoweledge about reasoning and confidence.My son has improved his vocabulary ...

Manish Kumar

10th

My experience with Abhyas is very good. I have learnt many things here like vedic maths and reasoning also. Teachers here first take our doubts and then there are assignments to verify our weak poi...

Shivam Rana

7th

It has a great methodology. Students here can get analysis to their test quickly.We can learn easily through PPTs and the testing methods are good. We know that where we have to practice

Barkha Arora

10th

About Abhyas metholodology the teachers are very nice and hardworking toward students.The Centre Head Mrs Anu Sethi is also a brilliant teacher.Abhyas has taught me how to overcome problems and has...

Shreya Shrivastava

8th

I have spent a wonderful time in Abhyas academy. It has made my reasoning more apt, English more stronger and Maths an interesting subject for me. It has given me a habbit of self studying

Yatharthi Sharma

10th

My experience with Abhyas academy is very good. I did not think that my every subject coming here will be so strong. The main thing is that the online tests had made me learn here more things.

Hiya Gupta

8th

Being a parent, I saw my daughter improvement in her studies by seeing a good result in all day to day compititive exam TMO, NSO, IEO etc and as well as studies. I have got a fruitful result from m...

Prisha Gupta

8th

When I have not joined Abhyas Academy, my skills of solving maths problems were not clear. But, after joining it, my skills have been developed and my concepts of science and SST are very well. I a...

Sharandeep Singh

7th

It was good as the experience because as we had come here we had been improved in a such envirnment created here.Extra is taught which is beneficial for future.

Eshan Arora

8th

In terms of methodology I want to say that institute provides expert guidence and results oriented monitering supplements by requsite study material along with regular tests which help the students to improve their education skills.The techniques of providing education helps the students to asses...

Aman Kumar Shrivastava

10th

Usually we see institutes offering objective based learning which usually causes a lag behind in subjective examinations which is the pattern followed by schools. I think it is really a work of planning to make us students grab the advantages of modes of examination, Objective Subjective and Onli...

Anika Saxena

8th

Third consective year,my ward is in Abhyas with nice experience of admin and transport support.Educational standard of the institute recumbent at satisfactory level. One thing would live to bring in notice that last year study books was distributed after half of the session was over,though study ...