Find a value of θ, in the interval [0°, 90°) that satisfies each statement. Give answers in decimal degrees to six decimal places. sec θ = 1.2637891

검증된 단계별 안내

Recall the definition of secant in terms of cosine: \(\sec \theta = \frac{1}{\cos \theta}\). This means that \(\cos \theta = \frac{1}{\sec \theta}\).

Substitute the given value of \(\sec \theta\) into the equation: \(\cos \theta = \frac{1}{1.2637891}\).

Calculate the value of \(\cos \theta\) using the reciprocal of the given secant value (do not compute the final number here, just set up the expression).

Use the inverse cosine function to find \(\theta\): \(\theta = \cos^{-1}(\text{value from step 3})\). This will give \(\theta\) in radians or degrees depending on your calculator settings.

Since the problem asks for \(\theta\) in degrees within the interval \([0^\circ, 90^\circ)\), ensure your calculator is set to degrees and express the answer to six decimal places.

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주요 개념

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

Definition of Secant Function

The secant function, sec θ, is the reciprocal of the cosine function, defined as sec θ = 1/cos θ. Understanding this relationship allows you to convert the given secant value into a cosine value, which is often easier to work with when solving for θ.

Inverse Trigonometric Functions

To find the angle θ from a trigonometric value, you use inverse functions such as arccosine. Since sec θ = 1/cos θ, you first find cos θ and then apply the arccos function to determine θ within the specified interval.

Domain and Range Restrictions

The problem restricts θ to the interval [0°, 90°), which corresponds to the first quadrant where cosine values are positive. This restriction ensures a unique solution and guides the selection of the correct angle from the inverse cosine calculation.