Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.

√2 sin 3x - 1 = 0

검증된 단계별 안내

Start by isolating the sine term in the equation: \( \sqrt{2} \sin 3x - 1 = 0 \). Add 1 to both sides to get \( \sqrt{2} \sin 3x = 1 \).

Next, divide both sides by \( \sqrt{2} \) to solve for \( \sin 3x \): \( \sin 3x = \frac{1}{\sqrt{2}} \).

Recall that \( \sin \theta = \frac{1}{\sqrt{2}} \) corresponds to angles where \( \theta = \frac{\pi}{4} + 2k\pi \) or \( \theta = \frac{3\pi}{4} + 2k\pi \) for any integer \( k \). Here, \( \theta = 3x \).

Set up the two equations for \( 3x \): \( 3x = \frac{\pi}{4} + 2k\pi \) and \( 3x = \frac{3\pi}{4} + 2k\pi \), where \( k \) is any integer.

Solve each equation for \( x \) by dividing both sides by 3: \( x = \frac{\pi}{12} + \frac{2k\pi}{3} \) and \( x = \frac{3\pi}{12} + \frac{2k\pi}{3} \). Then, find all solutions within the desired interval, and convert to degrees if needed, rounding as specified.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

도움이 되었나요?

주요 개념

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

Solving Trigonometric Equations

This involves isolating the trigonometric function and finding all angle values that satisfy the equation within a given domain. Solutions often include general forms using periodicity, and exact or approximate values depending on the problem's requirements.

Angle Measures and Unit Conversion

Understanding the difference between radians and degrees is essential, as well as converting between them. Radians are based on the radius of a circle, while degrees divide a circle into 360 parts. Correct unit usage ensures accurate solutions and proper rounding.

Using Inverse Trigonometric Functions and Periodicity

Inverse sine (arcsin) helps find principal angle solutions, but because sine is periodic with period 2π (or 360°), all solutions must account for this repetition. Writing solutions with the least nonnegative angle measures requires adjusting for these periodic intervals.

Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.√2 sin 3x - 1 = 0

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Solving Trigonometric Equations

Angle Measures and Unit Conversion

Using Inverse Trigonometric Functions and Periodicity