Express $\tan \theta$ in terms of $\tan \frac{\theta}{3}$.
Answer (1)
Substitute x = 3 θ , so θ = 3 x .
Apply the triple angle formula: tan 3 x = 1 − 3 t a n 2 x 3 t a n x − t a n 3 x .
Substitute back to get tan θ = 1 − 3 t a n 2 3 θ 3 t a n 3 θ − t a n 3 3 θ .
The final expression is: tan θ = 1 − 3 tan 2 3 θ 3 tan 3 θ − tan 3 3 θ .
Explanation
Understanding the Problem We want to express tan θ in terms of tan 3 θ . This problem involves trigonometric identities, specifically the triple angle formula for tangent.
Substitution Let x = 3 θ . Then θ = 3 x . Our goal is to express tan 3 x in terms of tan x .
Applying the Triple Angle Formula Recall the triple angle formula for tangent: tan 3 x = 1 − 3 tan 2 x 3 tan x − tan 3 x . This formula directly relates the tangent of 3 x to the tangent of x .
Final Expression Now, substitute x = 3 θ back into the formula: tan θ = tan ( 3 ⋅ 3 θ ) = 1 − 3 tan 2 3 θ 3 tan 3 θ − tan 3 3 θ . This expresses tan θ in terms of tan 3 θ .
Examples
In physics, when analyzing the motion of a pendulum or the behavior of waves, you often encounter trigonometric functions. Being able to express tan θ in terms of tan 3 θ can simplify complex equations and make calculations easier. For example, if you know the tangent of an angle divided by three, this formula allows you to directly find the tangent of the original angle, which can be useful in determining velocities or forces in a system.
