Find all solutions of each equation. 3 sin θ + 5 = ﹣2 sin θ

검증된 단계별 안내

Start by writing down the given equation: \(3 \sin \theta + 5 = -2 \sin \theta\).

To isolate the sine term, add \(2 \sin \theta\) to both sides to combine like terms: \(3 \sin \theta + 2 \sin \theta + 5 = 0\).

Simplify the left side by combining the sine terms: \(5 \sin \theta + 5 = 0\).

Next, subtract 5 from both sides to isolate the sine term: \(5 \sin \theta = -5\).

Divide both sides by 5 to solve for \(\sin \theta\): \(\sin \theta = -1\).

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주요 개념

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

Solving Trigonometric Equations

This involves isolating the trigonometric function (like sin θ) and finding all angle values that satisfy the equation within a given domain. Solutions often require algebraic manipulation followed by using inverse trigonometric functions.

Properties of the Sine Function

The sine function is periodic with period 2π and ranges between -1 and 1. Understanding its behavior helps determine all possible solutions for sin θ, including those in different quadrants where sine has the same value.

Using Inverse Sine and General Solutions

After isolating sin θ, the inverse sine function (arcsin) gives a principal solution. To find all solutions, one must use the general solution formulas θ = arcsin(x) + 2nπ and θ = π - arcsin(x) + 2nπ, where n is any integer.