Use a calculator to approximate the value of each expression. Give answers to six decimal places. sec 222° 30'

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Convert the angle from degrees and minutes to decimal degrees. Since 30' means 30 minutes, and 1 degree = 60 minutes, calculate the decimal degrees as \(222 + \frac{30}{60}\).

Recall that \(\sec \theta = \frac{1}{\cos \theta}\). So, to find \(\sec 222^\circ 30'\), you first need to find \(\cos 222.5^\circ\) (the decimal degree equivalent).

Use a calculator set to degree mode to find the value of \(\cos 222.5^\circ\).

Calculate the secant by taking the reciprocal of the cosine value: \(\sec 222.5^\circ = \frac{1}{\cos 222.5^\circ}\).

Round your final answer to six decimal places as requested.

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주요 개념

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Understanding the Secant Function

The secant function, sec(θ), is the reciprocal of the cosine function, defined as sec(θ) = 1/cos(θ). To find secant values, you first calculate the cosine of the angle and then take its reciprocal. This relationship is fundamental when working with trigonometric expressions involving secant.

Converting Degrees and Minutes to Decimal Degrees

Angles given in degrees and minutes must be converted to decimal degrees for calculator use. Since 1 minute equals 1/60 of a degree, convert 30' to 0.5°, so 222° 30' becomes 222.5°. This conversion ensures accurate input for trigonometric calculations on calculators.

Using a Calculator for Trigonometric Approximations

Calculators typically require angles in decimal degrees or radians to compute trigonometric functions. After converting the angle, input it correctly, ensure the calculator is in degree mode, compute cos(θ), then find sec(θ) by taking the reciprocal. Finally, round the result to six decimal places as requested.