QuestionFind the decimal approximation of $\sec 49^{\circ} 42^{\prime}$ rounded to seven decimal places.

Studdy Solution

STEP 1

Assumptions1. We are asked to find the decimal approximation of the secant function for the angle $49^{\circ}42^{\prime}$ . . The secant function is the reciprocal of the cosine function, i.e., $\sec x = \frac{1}{\cos x}$ .
3. The calculator is set to degree mode since the angle is given in degrees and minutes.

STEP 2

First, we need to convert the angle from degrees and minutes to decimal degrees. We can do this by dividing the number of minutes by60 (since there are60 minutes in a degree) and adding this to the number of degrees.
$Angle = Degrees + \frac{Minutes}{60}$

STEP 3

Now, plug in the given values for the degrees and minutes to calculate the decimal degree.
$Angle =49 + \frac{42}{60}$

STEP 4

Calculate the decimal degree.
$Angle =49 + \frac{42}{60} =49.7^{\circ}$

STEP 5

Now that we have the angle in decimal degrees, we can find the secant of this angle. As mentioned before, the secant is the reciprocal of the cosine. So, we first find the cosine of the angle and then take its reciprocal.
$\sec x = \frac{1}{\cos x}$

STEP 6

Plug in the value for the angle to calculate the secant.
$\sec49.^{\circ} = \frac{1}{\cos49.^{\circ}}$

STEP 7

Use a calculator to find the cosine of the angle.
$\cos49.7^{\circ} \approx0.656059028$

STEP 8

Calculate the secant of the angle by taking the reciprocal of the cosine.
$\sec49.7^{\circ} = \frac{1}{0.656059028}$

STEP 9

Calculate the secant of the angle.
$\sec49.7^{\circ} \approx.524868618$ The secant of $49^{\circ}42^{\prime}$ is approximately.524868618.

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