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# How to find the exact value of $\sec \left( { - \dfrac{\pi }{3}} \right)$ ? Verified
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Hint: This problem is easy once we find the definition of sec. A secant of a curve is a line that intersects the curve at least two distinct points in geometry. Secant is derived from the Latin word secare, which means "to cut." A secant can intersect a circle at exactly two points in the case of a circle.

Complete step by step solution:
$\sec = \dfrac{1}{{\cos }}$
Let's start by converting to degrees from radians. The conversion for radians to degrees
$\dfrac{{180}}{\pi }$.
$\dfrac{{180}}{\pi } \times \left( { - \dfrac{\pi }{3}} \right) \\ = \left( { - {{60}^\circ }} \right) \\$
To make this a positive angle, we must subtract 60 from 360, giving us ${300^\circ }$ .
This is a unique perspective because it provides us with a precise response. However, we must first determine the reference angle before adding our special triangle. A reference angle is the angle between the terminal side of $\theta$ to the x axis . It should always satisfy the interval ${0^\circ } \leqslant \beta < {90^\circ }$ . The nearest x axis interception of ${300^\circ }$ is at ${360^\circ }$ . After subtracting, we arrive at a reference angle of ${60^\circ }$ . We use the $30 - 60 - 90$ , 1 , $\sqrt 3$ , 2 .
Since 60 is greater than 30 , and ${60^\circ }$ is the reference angle , this indicates that the side opposite our reference angle has been measured $\sqrt 3$ . The hypotenuse is always the longest, with a length of 2. As a result, we can deduce that the adjacent side is 1.
Applying the definition of cos:
adjacent/hypotenuse $=$ $- \dfrac{1}{2}$ (cos is negative in quadrant IV)
Substituting into sec .
1/(adjacent/hypotenuse) $=$ hypotenuse/adjacent $= - 2$
Therefore, $\sec \left( { - \dfrac{\pi }{3}} \right) = - 2$.

Note:
Sine, cosine, and tangent are the three most common trigonometric ratios. However, even though they are rarely used, there are three additional ratios: secant, cosecant, and cotangent. Since they can be conveniently calculated using the three key ratios, most calculators don't even have a button for them.