Solve each equation for x.
y = 1/2 tan (3x + 2), for x in [-2/3 - π/6, -2/3 + π/6]

검증된 단계별 안내

Start by isolating the tangent function: multiply both sides of the equation by 2 to get \( \tan(3x + 2) = 2y \).

Next, take the inverse tangent (arctan) of both sides to solve for the angle: \( 3x + 2 = \arctan(2y) \).

Subtract 2 from both sides to isolate the term with x: \( 3x = \arctan(2y) - 2 \).

Divide both sides by 3 to solve for x: \( x = \frac{\arctan(2y) - 2}{3} \).

Ensure that the solution for x falls within the given interval \([-\frac{2}{3} - \frac{\pi}{6}, -\frac{2}{3} + \frac{\pi}{6}]\) by checking the calculated x values.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Tangent Function

The tangent function, denoted as tan(x), is a fundamental trigonometric function defined as the ratio of the opposite side to the adjacent side in a right triangle. It is periodic with a period of π, meaning it repeats its values every π radians. Understanding the properties of the tangent function, including its asymptotes and behavior, is crucial for solving equations involving tan.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arctan, are used to find angles when the value of a trigonometric function is known. For example, if y = tan(x), then x = arctan(y). These functions are essential for solving equations where the variable is inside a trigonometric function, allowing us to isolate the angle and find the corresponding x values.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. In this context, the interval [-2/3 - π/6, -2/3 + π/6] specifies the domain within which we are looking for solutions for x. Understanding how to interpret and work within specified intervals is important for determining valid solutions to trigonometric equations.

Solve each equation for x. y = 1/2 tan (3x + 2), for x in [-2/3 - π/6, -2/3 + π/6]