Find a value of θ, in the interval [0°, 90°) that satisfies each statement. Give answers in decimal degrees to six decimal places. csc θ = 9.5670466

검증된 단계별 안내

Recall the definition of the cosecant function: \(\csc \theta = \frac{1}{\sin \theta}\). This means that \(\sin \theta = \frac{1}{\csc \theta}\).

Substitute the given value of \(\csc \theta\) into the equation: \(\sin \theta = \frac{1}{9.5670466}\).

Calculate the value of \(\sin \theta\) using the reciprocal of \(9.5670466\) (do not compute the final decimal here, just set up the expression).

Use the inverse sine function to find \(\theta\): \(\theta = \sin^{-1} \left( \frac{1}{9.5670466} \right)\).

Since \(\theta\) must be in the interval \([0^\circ, 90^\circ)\), the principal value from the inverse sine function will be the solution. Express \(\theta\) in decimal degrees to six decimal places.

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주요 개념

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

Reciprocal Trigonometric Functions

The cosecant function (csc θ) is the reciprocal of the sine function, defined as csc θ = 1/sin θ. Understanding this relationship allows you to convert between csc θ and sin θ, which is essential for solving equations involving csc θ.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsin, are used to find the angle θ when the value of a trigonometric function is known. After finding sin θ from csc θ, you use arcsin to determine θ within the specified interval.

Domain and Range Restrictions

When solving trigonometric equations, it is important to consider the domain and range of the angle θ. Here, θ is restricted to [0°, 90°), meaning the solution must be an acute angle, which affects the choice of the correct inverse function value.