Ch. 3 - Trigonometric Identities and Equations

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Blitzer 3rd Edition

Ch. 3 - Trigonometric Identities and Equations

Problema 3.5.29

Capitolo 3, Problema 3.5.29

Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). tan 3x = (√3)/3

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Start by writing down the given equation: $\tan 3x = \frac{\sqrt{3}}{3}$.

Recall the values of tangent for common angles. Since $\tan \theta = \frac{\sqrt{3}}{3}$, identify the reference angle $\theta$ where this is true. This corresponds to $\theta = \frac{\pi}{6}$ (or 30 degrees).

Set up the general solution for \$3x$ using the periodicity of the tangent function, which has period $\pi$. So, $3x = \frac{\pi}{6} + k\pi$, where $k$ is any integer.

Solve for $x$ by dividing both sides by 3: $x = \frac{\pi}{18} + \frac{k\pi}{3}$.

Find all values of $x$ in the interval $[0, 2\pi)$ by substituting integer values of $k$ such that $x$ remains within this interval.

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Multiple-Angle Trigonometric Equations

These are equations where the trigonometric function's argument is a multiple of the variable, such as tan(3x). Solving them requires understanding how to handle the periodicity and multiple solutions within a given interval, often by dividing the interval accordingly.

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Properties and Periodicity of the Tangent Function

The tangent function has a period of π, meaning tan(θ) = tan(θ + nπ) for any integer n. This periodicity is crucial when solving equations like tan(3x) = value, as it helps find all solutions within the specified interval by considering all possible angle shifts.

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Solving Trigonometric Equations on a Restricted Interval

When solving trig equations on [0, 2π), it is important to find all solutions within this range. For multiple-angle equations, the variable's domain is adjusted accordingly (e.g., 3x in [0, 6π)), and solutions are then translated back to x by dividing by the multiple.

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