Solve the following equations.{Use of Tech} sin⁡2θ=15,0\u003cθ\u003cπ2\sin2\theta=\frac15,0\lt{\theta}\lt{\frac{\pi}{2}}sin2θ=51,0\u003cθ\u003c2π

Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 1 - Functions

문제 45

1장, 문제 45

Solve the following equations.

{Use of Tech} $$\sin$2$\theta$=$\frac$15,0$\lt{\theta}\]\lt{\frac{\pi}{2}$}$

검증된 단계별 안내

First, recognize that the equation given is $ \sin 2\theta = \frac{1}{5} $. This is a trigonometric equation involving the sine function.

Next, use the inverse sine function to solve for $ 2\theta $. This gives $ 2\theta = \arcsin\left(\frac{1}{5}\right) $.

Since the sine function is periodic with a period of $ 2\pi $, consider the general solution for $ 2\theta $, which is $ 2\theta = \arcsin\left(\frac{1}{5}\right) + 2k\pi $ or $ 2\theta = \pi - \arcsin\left(\frac{1}{5}\right) + 2k\pi $, where $ k $ is an integer.

Now, solve for $ \theta $ by dividing the entire equation by 2, giving $ \theta = \frac{1}{2}\arcsin\left(\frac{1}{5}\right) + k\pi $ or $ \theta = \frac{1}{2}(\pi - \arcsin\left(\frac{1}{5}\right)) + k\pi $.

Finally, apply the constraint $ 0 < \theta < \frac{\pi}{2} $ to find the specific values of $ \theta $ that satisfy the original equation within the given interval. Check each possible solution to ensure it falls within this range.

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주요 개념

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

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. The sine function, specifically, gives the ratio of the length of the opposite side to the hypotenuse in a right triangle. Understanding these functions is crucial for solving equations involving angles, particularly in the context of periodic functions and their properties.

Double Angle Formulas

Double angle formulas are identities that express trigonometric functions of double angles in terms of single angles. For example, the sine double angle formula states that sin(2θ) = 2sin(θ)cos(θ). This concept is essential for simplifying and solving equations that involve angles multiplied by two, as seen in the given equation.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. In the context of the given problem, the interval 0 < θ < π/2 indicates that θ must be a positive angle less than 90 degrees. Understanding interval notation is important for determining the valid solutions to trigonometric equations, ensuring that the solutions fall within specified bounds.

Solve the following equations.{Use of Tech} sin2θ=15,0<θ<π2\(\sin\)2\(\theta\)=\(\frac\)15,0\(\lt{\theta}\]\lt{\frac{\pi}{2}\)}sin2θ=51,0<θ<2π

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Trigonometric Functions

Double Angle Formulas

Interval Notation

Solve the following equations.

{Use of Tech} $\(\sin\)2\(\theta\)=\(\frac\)15,0\(\lt{\theta}\]\lt{\frac{\pi}{2}\)}$