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QuestionFind the value of sec⁡3690∘\sec 3690^{\circ} or state if it is undefined.

Studdy Solution

STEP 1

Assumptions1. We are asked to find the value of the secant function at3690 degrees. . The secant function is undefined when the cosine function is zero, i.e., at odd multiples of90 degrees.

STEP 2

First, we need to reduce the angle to a value within the standard range of0 to360 degrees. We can do this by using the property that the trigonometric functions are periodic with a period of360 degrees.
θ=3690∘mod  360∘\theta =3690^{\circ} \mod360^{\circ}

STEP 3

Calculate the reduced angle.
θ=3690∘mod  360∘=90∘\theta =3690^{\circ} \mod360^{\circ} =90^{\circ}

STEP 4

Now, we can find the secant of the reduced angle. The secant function is the reciprocal of the cosine function.
sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}

STEP 5

Plug in the value of the reduced angle into the cosine function.
sec⁡90∘=1cos⁡90∘\sec90^{\circ} = \frac{1}{\cos90^{\circ}}

STEP 6

The cosine of90 degrees is zero.
cos⁡90∘=0\cos90^{\circ} =0

STEP 7

Substitute the value of the cosine function into the secant function.
sec⁡90∘=10\sec90^{\circ} = \frac{1}{0}

STEP 8

The division by zero is undefined in mathematics. Therefore, the secant of90 degrees is undefined.
The secant of3690 degrees is undefined.

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