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
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Question : 1

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 )

Right Option : D
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Question : 2

Find the value of

frac{tan;40^0}{cot;50^0}+(sin^2;20^0+sin^2;70^0)+ tan;5^0tan;10^0tan;30^0tan;80^0tan;85^0

Right Option : C
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Explanation
Question : 3

Evaluate sin^2;A+cos^2;A+3=

Right Option : B
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Explanation
 
 


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