Solve the equation sin 2Θ = 1, for 0 ≤ Θ \u003c 2π .

Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Briggs 3rd Edition

Ch. 1 - Functions

문제 10.

1장, 문제 10.

Solve the equation sin 2Θ = 1, for 0 ≤ Θ < 2π .

검증된 단계별 안내

Recognize that \( \sin 2\Theta = 1 \) implies that the angle \( 2\Theta \) is at a point where the sine function equals 1.

Recall that the sine function equals 1 at \( \frac{\pi}{2} + 2k\pi \), where \( k \) is an integer.

Set \( 2\Theta = \frac{\pi}{2} + 2k\pi \) and solve for \( \Theta \) by dividing both sides by 2.

This gives \( \Theta = \frac{\pi}{4} + k\pi \).

Determine the values of \( k \) such that \( 0 \leq \Theta < 2\pi \) to find the solutions within the given interval.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate angles to ratios of sides in right triangles. The sine function, specifically, gives the ratio of the length of the opposite side to the hypotenuse. Understanding these functions is crucial for solving equations involving angles, as they describe periodic behaviors and relationships in geometry.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to determine the angle that corresponds to a given trigonometric ratio. For example, if we know sin(Θ) = 1, we can use the inverse sine function to find the angle Θ. This concept is essential for solving equations where the angle is the unknown variable.

Periodic Nature of Sine Function

The sine function is periodic, meaning it repeats its values in regular intervals. Specifically, sin(Θ) has a period of 2π, which means that sin(Θ) = sin(Θ + 2πk) for any integer k. This property is important when solving equations like sin(2Θ) = 1, as it allows us to find multiple solutions within a specified interval.