tan3θ = 3tanθ - tan^3θ/ 1-3tan^2θ
To simplify the trigonometric expression tan(3θ) = (3tan(θ) - tan^3(θ))/ (1 - 3tan^2(θ)), we can use the trigonometric identity for the tangent of a triple angle:
tan(3θ) = (3tan(θ) - tan^3(θ))/(1 - 3tan^2(θ))
Using another trigonometric identity, we have:
tan(3θ) = (tan(θ) * (3 - tan^2(θ)))/(1 - 3tan^2(θ))
Now, let's simplify further:
tan(3θ) = (3tan(θ) - tan^3(θ))/(1 - 3tan^2(θ))
tan(3θ) = (tan(θ) * (3 - tan^2(θ)))/(1 - 3tan^2(θ))
tan(3θ) = (tan(θ) * (3(1 - tan^2(θ))))/((1 - tan^2(θ))(1 + tan^2(θ)))
The (1 - tan^2(θ)) term in the numerator and the denominator will cancel out, leaving us with:
tan(3θ) = 3tan(θ)/(1 + tan^2(θ))
So, the simplified expression is tan(3θ) = 3tan(θ)/(1 + tan^2(θ)).
To simplify the given expression tan(3θ) = 3tan(θ) - tan^3(θ) / (1 - 3tan^2(θ)), we can use trigonometric identities to rewrite and simplify the equation. First, let's factor out a common factor of tan(θ) from the numerator:
tan(3θ) = tan(θ)(3 - tan^2(θ)) / (1 - 3tan^2(θ))
Next, we can write tan(3θ) as sin(3θ) / cos(3θ) using the identity tan(θ) = sin(θ) / cos(θ):
sin(3θ) / cos(3θ) = tan(θ)(3 - tan^2(θ)) / (1 - 3tan^2(θ))
To simplify further, we can use the double-angle formula for sine and cosine:
sin(3θ) = 3sin(θ) - 4sin^3(θ)
cos(3θ) = 4cos^3(θ) - 3cos(θ)
Substituting these values back into the equation, we get:
(3sin(θ) - 4sin^3(θ)) / (4cos^3(θ) - 3cos(θ)) = tan(θ)(3 - tan^2(θ)) / (1 - 3tan^2(θ))
Now, let's manipulate the equation to get a common denominator:
(3sin(θ) - 4sin^3(θ)) / (4cos^3(θ) - 3cos(θ)) = tan(θ)(3 - tan^2(θ)) / (1 - 3tan^2(θ))
Multiplying both sides of the equation by (4cos^3(θ) - 3cos(θ)), we get:
(3sin(θ) - 4sin^3(θ)) = tan(θ)(3 - tan^2(θ))(4cos^3(θ) - 3cos(θ)) / (1 - 3tan^2(θ))
At this point, we have simplified the equation as much as possible using trigonometric identities. The result is a long expression involving trigonometric functions. If you need further simplification or have specific values for θ that need to be substituted, please provide those values for a more specific answer.
tan 3Ø = tan (2Ø + Ø)
= (tan 2Ø + tanØ)/(1 - tan 2Ø tanØ)
and tan 2Ø = 2tanØ/(1 - tan^2 Ø)
let x = tanØ for ease of typing
then tan 3Ø
= (tan 2Ø + x)/(1 - xtan 2Ø)
= ( (2x/(1 - x^2) + x)/(1 - x(2x/(1-x^2) )
= [ (2x + x - x^3)/(1 - x^2)]/[ (1 - x^2 - 2x^2)/(1 - x^2 ]
= (3x - x^3)/(1 - 3x^2)
= (3tanØ - tan^3 Ø)/(1 - 3tan^2 Ø)
= RS