Find a value of θ in the interval [0°, 90°) that satisfies each statement. Give answers in decimal degrees to six decimal places. See Example 2.
sin θ = 0.84802194

검증된 단계별 안내

Identify the given equation: \(\sin \theta = 0.84802194\) and the interval for \(\theta\) is \([0^\circ, 90^\circ)\).

Recall that the sine function is positive and increasing in the interval \([0^\circ, 90^\circ)\), so there will be exactly one solution in this range.

Use the inverse sine function (arcsin) to find \(\theta\): \(\theta = \arcsin(0.84802194)\).

Make sure your calculator is set to degree mode before calculating \(\arcsin(0.84802194)\) to get the angle in degrees.

Calculate the value of \(\theta\) and round it to six decimal places as required.

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주요 개념

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

Inverse Sine Function (Arcsin)

The inverse sine function, denoted as arcsin or sin⁻¹, is used to find the angle whose sine value is given. For a value y in the range [-1, 1], arcsin(y) returns an angle θ in the interval [-90°, 90°]. This is essential for determining θ when sin θ is known.

Domain and Range of Sine Function

The sine function outputs values between -1 and 1 for all real angles. When solving for θ in [0°, 90°), the sine function is positive and increasing, ensuring a unique solution for sin θ = 0.84802194 within this interval.

Decimal Degree Precision

Angles can be expressed in degrees with decimal precision for accuracy. Here, the solution requires θ to be given to six decimal places, which means careful use of a calculator or software to obtain and report the angle with high precision.

Find a value of θ in the interval [0°, 90°) that satisfies each statement. Give answers in decimal degrees to six decimal places. See Example 2. sin θ = 0.84802194