Express $\tan \theta$ in term of $\tan \frac{\theta}{3}$.
Answer (1)
Let x = 3 θ , so we want to express tan ( 3 x ) in terms of tan ( x ) .
Apply the triple angle formula for tangent: tan ( 3 x ) = 1 − 3 t a n 2 ( x ) 3 t a n ( x ) − t a n 3 ( x ) .
Substitute x = 3 θ into the formula.
The expression for tan θ in terms of tan 3 θ is 1 − 3 tan 2 ( 3 θ ) 3 tan ( 3 θ ) − tan 3 ( 3 θ ) .
Explanation
Problem Analysis and Setup We need to express tan ( θ ) in terms of tan ( 3 θ ) . Let's denote x = 3 θ , which means θ = 3 x . Our goal is to find an expression for tan ( 3 x ) using tan ( x ) .
Applying the Triple Angle Formula We will use the triple angle formula for tangent, which states: 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 .
Substitution and Final Expression Now, we substitute x = 3 θ back into the triple angle formula: tan ( θ ) = tan ( 3 ⋅ 3 θ ) = 1 − 3 tan 2 ( 3 θ ) 3 tan ( 3 θ ) − tan 3 ( 3 θ ) This gives us the expression for tan ( θ ) in terms of tan ( 3 θ ) .
Final Answer Therefore, the expression for tan θ in terms of tan 3 θ is: tan ( θ ) = 1 − 3 tan 2 ( 3 θ ) 3 tan ( 3 θ ) − tan 3 ( 3 θ )
Examples
Imagine you're designing a robotic arm that needs to rotate to a specific angle, but the motor only allows you to control smaller, fractional angles. Knowing how to express the tangent of a larger angle in terms of the tangent of a smaller angle (like dividing the angle by 3) helps you calculate the precise adjustments needed for the robot's movements. This is crucial in fields like robotics, where precise angular movements are essential for tasks such as manufacturing, surgery, or exploration.
