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 $frac{sin;65^0}{cos;25^0}+(sin^2;15^0+sin^2;75^0)+ sin;5^0 cos;10^0 tan;45^0cosec;80^0sec;85^0$
A. $frac{1}{sqrt3}$	B. 3	C. 1	D. None of these
Right Option : B
View Explanation

Question : 3

Find the value of $sin^2;10^0+sin^2; 30^0 +sin^2;60^0+sin^2;80^0$ .
A. 4	B. $frac{3}{2}$	C. 2	D. $frac{1}{4}$
Right Option : C
View Explanation

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Trigonometric Identity Involving Sine And Cosine

Trigonometric Identity Involving Sine And Cosine

Students / Parents Reviews [10]

Cheshta

Manish Kumar

Yatharthi Sharma

Ayan Ghosh

Shreya Shrivastava

Aman Kumar Shrivastava

Om Umang

Eshan Arora

Barkha Arora

Archit Segal