Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). 3θ __ cot -------- = ﹣√ 3 2

Verified step by step guidance

1. Recognize that $\cot \left(\frac{3\theta }{2}\right)=-\sqrt{3}$ implies $\frac{3\theta }{2}$ is an angle where the cotangent value is $-\sqrt{3}$.
2. Recall that $\cot (\theta )=-\sqrt{3}$ corresponds to angles where $\tan (\theta )=-\frac{1}{\sqrt{3}}$, which are $\theta =\frac{2\pi }{3}$ and $\theta =\frac{5\pi }{3}$ in the interval $[0,2\pi )$.
3. Set $\frac{3\theta }{2}=\frac{2\pi }{3}+k\pi$ and $\frac{3\theta }{2}=\frac{5\pi }{3}+k\pi$ for integer $k$, to account for the periodicity of the cotangent function.
4. Solve each equation for $\theta$ by multiplying both sides by $\frac{2}{3}$: $\theta =\frac{4\pi }{9}+\frac{2k\pi }{3}$ and $\theta =\frac{10\pi }{9}+\frac{2k\pi }{3}$.
5. Determine the values of $\theta$ within the interval $[0,2\pi )$ by substituting integer values for $k$ and checking if the resulting $\theta$ falls within the specified range.

Key Concepts

Cotangent Function

The cotangent function, denoted as cot(θ), is the reciprocal of the tangent function, defined as cot(θ) = cos(θ)/sin(θ). It is important in solving trigonometric equations, especially those involving angles, as it relates the angles to the ratios of the sides of a right triangle. Understanding how to manipulate and interpret cotangent values is essential for solving equations like the one presented.

Multiple Angle Equations

Multiple angle equations involve trigonometric functions of angles that are multiples of a variable, such as 3θ in this case. These equations can yield multiple solutions within a specified interval, necessitating the use of identities and periodic properties of trigonometric functions. Recognizing how to apply these concepts helps in finding all possible solutions within the given range.

Interval Notation

Interval notation specifies the range of values for which a solution is valid. In this case, the interval [0, 2π) indicates that solutions should be found starting from 0 up to, but not including, 2π. Understanding how to interpret and apply interval notation is crucial for ensuring that all solutions are correctly identified and fall within the specified range.