Understanding the Secant Function in Trigonometry and Calculating sec 30°

sec 30°

In mathematics, sec is the abbreviation for the secant function

In mathematics, sec is the abbreviation for the secant function. The secant function is a trigonometric function that denotes the ratio of the hypotenuse to the adjacent side in a right triangle. It is defined as the reciprocal of the cosine function.

To evaluate sec 30°, we can use the properties of the unit circle or trigonometric identities. The unit circle is a circle with a radius of 1 centered at the origin (0,0) in a coordinate plane.

At 30° on the unit circle, we can draw a right triangle with an angle of 30° at one of the vertices. The adjacent side in this triangle is equal to 1, as it corresponds to the x-coordinate on the unit circle.

Using the Pythagorean theorem, we can find the hypotenuse of the triangle. The opposite side is equal to half the hypotenuse, as it corresponds to the y-coordinate on the unit circle. Let’s call the hypotenuse H.

From the Pythagorean theorem, we have:
(1)^2 + (H/2)^2 = H^2

Simplifying the equation, we get:
1 + (H^2)/4 = H^2
4 + H^2 = 4H^2
3H^2 = 4
H^2 = 4/3
H = √(4/3) = 2/√3 = 2√3/3

Now, to calculate sec 30°, we take the reciprocal of the cosine function. The cosine of 30° is equal to the adjacent side divided by the hypotenuse, which is 1/(2√3/3). Simplifying this fraction, we get:
1/(2√3/3) = 1 * (3/2√3) = 3/2√3

Finally, taking the reciprocal, we have:
sec 30° = 2√3/3

Therefore, sec 30° is equal to 2√3/3.

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