Trigonometric Identities
Trigonometric identities are equivalent like that of trigonometric functions which are correct for all distinct value of the occurring variables. Those identities involves certain functions of one or more angles..Trigonometric function circumstances the connection between side lengths and internal angles of a right triangle. In this online table help we can get more help like various formulas like pythagorean,reciprocal,product,sum,sine law,cosine law etc...nd the following are the trigonometric Identities exposed with formulas
1. Trigonometric Identities for Pythagorean :
`sin2 a + cos2 a = 1`
`1 + tan2 a = sec2 a`
`1 + cot2 a = csc2 a`
2. Trigonometric Identities for Reciprocal:
`csc a = 1/ sin a`
`sec a = 1/ cos a`
`cot a = 1/ tan a`
`sin a = 1/csc a`
`cos a = 1/sec a`
`tan a = 1/cot a`
3. Trigonometric Identities for Quotient:
`tan a = sin a/cos a`
`cot a= cos a/sin a`
4. Trigonometric Identities for Co-Function:
`sin(pi/2- a) = cos a`
`tan(pi/2- a) = cot a`
`sec(pi/2- a) = csc a`
`cos(pi/2- a) = sin a`
`cot(pi/2- a) = tan a`
`csc(pi/2- a) = sec a`
5.Trigonometric Identities for Even-Odd:
`cos(-a) = cos a`
`sin(-a) = -sin a`
`tan(-a) = -tan a`
`sec(-a) = sec a`
`csc(-a) = -csc a`
`cot(-a) = -cot a`
6. Trigonometric Identities for Sum-Difference Formulas:
`sin(a + b) = sin a cos b + cos a sin b`
`tan(a + b) = (tan a+tan b)/(tan a tan b)`
`sin(a - b) = sin a cos b - cos a sin b`
`cos(a + b) = cos a cos b - sin a sin b`
`tan(a - b) = (tan a-tan b)/(1+tan a tan b)`
`cos(a - b) = cos a cos b + sin a sin b`
7. Trigonometric Identities for Double Angle Formulas:
`sin(2a) = 2 sin a cos a`
`cos(2a) = cos^2 a - sin^2 a = 2 cos^2 a - 1 = 1 - 2 sin^2 a`
`tan(2a) = (2 tana)/(1-tan2 a)`
9.Trigonometric Identities for Sum-to-Product Formulas:
`sin a + sin b = 2 sin((a+b)/2 ) cos( (a-b)/2 )`
`sin a - sin b = 2 cos((a+b)/2 ) sin((a-b)/2 )`
`cos a + cos b = 2 cos((a+b)/2) cos(( a-b)/2 )`
`cos a - cos b = -2 sin((a+b)/2 ) sin((a-b)/2 )`
10.Trigonometric Identities for The Law of Sines:
`(sinA)/(a) = sinB/(b) = sinC/(c)`
11.Trigonometric Identities for The Law of Cosines :
`a^2 = b^2 + c^2 - 2bc cosA`
`b^2 = a^2 + c^2 - 2ac cosB`
`c^2 = a^2 + b^2 - 2ab cosC`
Below are the examples on trigonometric identities -
Example 1:
Prove the trigonometric identities : `sin y + sin y cot2 y = csc y`
`sin y + sin y cot2 y`
Step1:
By simplifying the given equation we get
= `sin y(1+ cot2 y)`
Step2:
We know that `csc2y=1+cot2y`
`= sin y( csc2 y )`
Step3:
We can write `csc2 y=1/(sin2y)`
`= (sin y )[(1)/(sin2 y)]`
Step4:
By crossing the values we get,
`=1/sin y`
Step5:
we know that `1/sin y= csc y`
`=csc y`
Hence the identity `sin y + sin y cot2 y = csc y` is proved.
Example 2:
Prove the identity:` tan a/sin a=sec a`
Step1:
Simplify the terms in sin and cos.Then simplify
`tan a . 1/(sin a)`
Step2:
The` tan a` can be put as `sin a/cos a`
`=(sin a)/(cos a). 1/(sin a)`
Step3:
Cross out the values we get,
`=1/(cos a)`
Step4:
We know that `1/cos a = sec a`
`=sec a`
Hence `tan a/sin a=sec a` is proved.
Below are the examples on trigonometric identities -
Example 1:
Prove the trigonometric identities : `sin y + sin y cot2 y = csc y`
`sin y + sin y cot2 y`
Step1:
By simplifying the given equation we get
= `sin y(1+ cot2 y)`
Step2:
We know that `csc2y=1+cot2y`
`= sin y( csc2 y )`
Step3:
We can write `csc2 y=1/(sin2y)`
`= (sin y )[(1)/(sin2 y)]`
Step4:
By crossing the values we get,
`=1/sin y`
Step5:
we know that `1/sin y= csc y`
`=csc y`
Hence the identity `sin y + sin y cot2 y = csc y` is proved.
Example 2:
Prove the identity:` tan a/sin a=sec a`
Step1:
Simplify the terms in sin and cos.Then simplify
`tan a . 1/(sin a)`
Step2:
The` tan a` can be put as `sin a/cos a`
`=(sin a)/(cos a). 1/(sin a)`
Step3:
Cross out the values we get,
`=1/(cos a)`
Step4:
We know that `1/cos a = sec a`
`=sec a`
Hence `tan a/sin a=sec a` is proved.
