Find two values of θ, 0 ≤ θ \u003c 2𝜋, that satisfy each equation. sin θ = - √2/2

Ch. 1 - Angles and the Trigonometric Functions

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Blitzer 3rd Edition

Ch. 1 - Angles and the Trigonometric Functions

문제 101

1장, 문제 101

Find two values of θ, 0 ≤ θ < 2𝜋, that satisfy each equation. sin θ = - √2/2

검증된 단계별 안내

Recognize that the equation is \( \sin \theta = -\frac{\sqrt{2}}{2} \). This means we are looking for angles \( \theta \) where the sine value is negative and equal to \( -\frac{\sqrt{2}}{2} \).

Recall the reference angle where \( \sin \theta = \frac{\sqrt{2}}{2} \) is \( \frac{\pi}{4} \). Since sine is negative, \( \theta \) must be in the third or fourth quadrants where sine values are negative.

Use the unit circle to find the two angles in the interval \( 0 \leq \theta < 2\pi \) that correspond to the sine value \( -\frac{\sqrt{2}}{2} \). These angles are \( \pi + \frac{\pi}{4} \) and \( 2\pi - \frac{\pi}{4} \).

Write the two solutions explicitly as \( \theta = \pi + \frac{\pi}{4} \) and \( \theta = 2\pi - \frac{\pi}{4} \).

Verify that both values lie within the given interval \( 0 \leq \theta < 2\pi \) and satisfy the original equation.

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주요 개념

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

Unit Circle and Angle Measurement

The unit circle is a circle with radius 1 centered at the origin, used to define trigonometric functions for all angles. Angles are measured in radians from 0 to 2π for one full rotation, helping locate sine values on the circle.

Sine Function and Its Range

The sine function gives the y-coordinate of a point on the unit circle corresponding to an angle θ. Its values range between -1 and 1, so any sine value must lie within this interval to have real solutions.

Solving Trigonometric Equations

To solve equations like sin θ = -√2/2, identify all angles θ within the given interval where sine equals that value. Since sine is negative in the third and fourth quadrants, find corresponding reference angles and adjust for these quadrants.